Hydrogen-atom wave functions nodes

Fig. 3 shows a qualitative graphical representation of hydrogen-like wave functions for one-electron atoms which have to be replaced for many-electron atoms at least by Slater-type 107) analytical wave functions ifnlm (1) which are approximate as they contain no nodes in the radial part R ,. 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

FIGURE 5.4 Four representations of hydrogen s orbitals, (a) A contour plot of the wave function amplitude for a hydrogen atom in its Is, 2s, and 3s states. The contours identify points at which i//takes on 0.05, 0.1, 0.3, 0.5, 0.7, and 0.9 of its maximum value. Contours with positive phase are shown in red those with negative phase are shown in blue. Nodal contours, where the amplitude of the wave function is zero, are shown in black. They are connected to the nodes in the lower plots by the vertical green lines, (b) The radial wave functions plotted against distance from the nucleus, r. (c) The radial probability density, equal to the square of the radial wave function multiplied by 1. (d) The "size" of the orbitals, as represented by spheres whose radius is the distance at which the probability falls to 0.05 of its maximum value. 

The separability or factorization of the wave functions of the hydrogen atom, in the respective coordinates, allows us to understand that their nodes correspond to conoidal surfaces or meridian planes. The latter become natural boundaries of confinement for the hydrogen atom, as natural extensions of Levine s plane [31], anticipating their discussion in Section 4. 

Fig. 4.16. A graphic representation of the 2-D hamonic oscillator wave function (isolinesj. Panels a) through ( i) show the wave functions labeled by a pair of oscillation quantum numbers (ui, V2). The higher the energy, the la ger the number of node planes. A reader acquainted with the wave functions of the hydrogen atom will easily recognize a striking resemblance between these figures and the orbitals.

Fig. 12.6. Polarization of the hydrogen atom in an electric field. The wave functions for (al the unperturbed atom (bl the atom in the electric field (a.u.) = (0.1, 0,0) are shown. As we can see, there are differences in the corresponding electronic density distributions in the second case, the wave function is deformed toward the anode (i.e., leftward). Note that the wave function is less deformed in the region close to the nucleus than in its left or right neighborhood. This is a consequence of the fact that the deformation is made by the —0.1986(2p., ) function. Its main role is to subtract on the right and add on the left, and the smallest changes are at the nucleus because 2px has its node there.

When two hydrogen l5 orbitals overlap out of phase with each other, an antibonding molecular orbital results (Figure 2-7). The two wave functions have opposite signs, so they tend to cancel out where they overlap. The result is a node (actually a nodal plane) separating the two atoms. The presence of a node separating the two nuclei usually indicates that the orbital is antibonding. 

Calculate the finite value of r, in terms of Uq, at which the node occurs in the wave function of the 2s orbital of a hydrogen atom. 

The Bohr atomic model, which describes an electron as an orbiting particle, is well known to fail for all atoms other than hydrogen. Maxima in the optimization function should therefore not be interpreted as orbits but rather as the nodes of a spherical standing wave in line with the periodic table of the elements. 

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