c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come
260 WAVE MECHANICS OF SIMPLE SYSTEMSFIGURE 16.7 Reduced degeneracy in the energy levels for hydrogen-like atoms. Some of
the degeneracy of Fig. 16.6 has been lost.maximum probability. Consequently, the energy of an electron drifting about in a
distantporbital is less negative than that of a tightly heldselectron. This causes the
spdegeneracy of the hydrogen atom to be broken up in beryllium, boron, carbon, and
the heavier elements. The 2slevel is lower than the 2plevels in beryllium because of
its greater average radial distance from the nucleus and because ofsshielding.
We see this when we traverse the series H→He→Li→Be→B in the atomic
table. The 1slevel readily accommodates H and He, but the electron in Li is excluded
from the 1slevel which is “filled” by two electrons.^1 Itcouldgo into either the 2sor
one of 2plevels, but in fact it goes into the 2slevel which has a lower energy. This
preference is even more remarkable in beryllium. The valence electron in Be has good
reason to go into the 2porbital to escape charge and spin correlation with the electron
already in the 2sorbital, but it refuses this safe haven and goes into the lower energy
2 salternative. We know from ionization potentials that the electron in Be ionizes
from the 2sorbital, not the 2p, and that Be is a typical metal, not as metallic as Li
perhaps, but not a metalloid like boron. Atoms with orbital configurations similar to
hydrogen but with reduced degeneracy are calledhydrogen-likeatoms (Fig. 16.7).16.8.1 Higher Exact Solutions for the Hydrogen Atom
The first six eigenfunctions of the hydrogen atom are shown in Table 16.1. The
geometric parametera= 52 .9pm= 5. 29 × 10 −^11 mis1/α, the first orbital radius
in Bohr theory. The integerZis the atomic number. It is 1 for hydrogen but 2 or 3
for the higher hydrogen-like atoms shown here. The orbitals fall into two different
groups,sandp. More complete listings included, f, g,...orbitals of increasing
complexity. Note that the terms function and orbital are used as synonyms. Thes
eigenfunctions are 1s,2s, and 3s. They do not contain sin or cos parts because they
do not depend upon angular location. They arespherically symmetrical. Table 16.2
shows the orbitals reduced to their simple functional form. Let us concentrate on the
simple polynomial form of thesorbitals by ignoring the premultiplying constants.
The first orbital is simply a negative exponential. From the monotonic decrease of
e(−r, we expect zero roots for the 1sfunction, but from the polynomials( 1 −r)and
1 −r+r^2)
we expect two and three roots, respectively. A Mathcad©Cgraph of the
3 sradial wave function in Fig. 16.8 shows that the polynomial part produces two
nodes and one asymptotic approach when the function is plotted on the vertical axis
againstras the independent variable.(^1) The universal restriction of two electrons with opposed spins per orbital is called the Pauli exclusion
principle.