Page 184 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.4 Atomic orbitals 175

unbound or scattering states and are useful in the study of electron±ion
collisions and scattering phenomena. In view of the complexity of the analysis
for obtaining the eigenfunctions and eigenvalues of equation (6.17) for positive
E and the unimportance of these quantities in most problems of chemical
interest, we do not consider this case any further.

In®nite nuclear mass
The energy levels E n and the radial functions R nl (r) depend on the reduced
mass ì of the two-particle system
m N m e m e
ì ˆ ˆ
m N ‡ m e m e
1 ‡
m N
where m N is the nuclear mass and m e is the electronic mass. The value of m e is
9:109 39 3 10 ÿ31 kg. For hydrogen, the nuclear mass is the protonic mass,
1:672 62 3 10 ÿ27 kg, so that ì is 9:1044 3 10 ÿ31 kg. For heavier hydrogen-like
atoms, the nuclear mass is, of course, greater than the protonic mass. In the
limit m N !1, the reduced mass and the electronic mass are the same. In the
classical two-particle problem of Section 6.1, this limit corresponds to the
nucleus remaining at a ®xed point in space.
In most applications, the reduced mass is suf®ciently close in value to the
electronic mass m e that it is customary to replace ì in the expressions for the
2
2
energy levels and wave functions by m e . The parameter a ì ˆ " =ìe9 is
2
2
thereby replaced by a 0 ˆ " =m e e9 . The quantity a 0 is, according to the earlier
Bohr theory, the radius of the circular orbit of the electron in the ground state
of the hydrogen atom (Z ˆ 1) with a stationary nucleus. Except in Section 6.5,
where this substitution is not appropriate, we replace ì by m e and a ì by a 0 in
the remainder of this book.

6.4 Atomic orbitals
We have shown that the simultaneous eigenfunctions ø(r, è, j) of the opera-
^
^ ^ 2
tors H, L , and L z have the form
ø nlm (r, è, j) ˆjnlmiˆ R nl (r)Y lm (è, j) (6:56)
where for convenience we have introduced the Dirac notation. The radial
functions R nl (r) and the spherical harmonics Y lm (è, j) are listed in Tables 6.1
and 5.1, respectively. These eigenfunctions depend on the three quantum
numbers n, l, and m. The integer n is called the principal or total quantum
number and determines the energy of the atom. The azimuthal quantum
number l determines the total angular momentum of the electron, while the