Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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ˆ 1
12
(4ÿ r)reÿr=2
S41 ˆ

3
40
r
r
d
dr
ÿ r
2
‡ 4
\ufffd \ufffd
1
12
(4ÿ r)reÿr=2
ˆ 1
8

30
p (20ÿ 10r‡ r2)reÿr=2
..
.
The functions S31, S41, . . . are automatically normalized as specified by
equation (6.25). The normalized eigenfunctions Snl(r) for l ˆ 2, 3, 4, . . . with
n > (l ‡ 1) are obtained by the same procedure.
A general formula for Snl involves the repeated application of B^k for
k ˆ l ‡ 2, l ‡ 3, . . . , nÿ 1, n to Sl‡1, l in equation (6.49). The raising operator
must be applied (nÿ l ÿ 1) times. The result is
Snl ˆ (bnl)ÿ1(bnÿ1, l)ÿ1 . . . (bl‡2, l)ÿ1 B^n B^nÿ1 . . . B^l‡2Sl‡1, l
ˆ (l ‡ 1)(2l ‡ 1)!
n(n‡ l)!(nÿ l ÿ 1)!(2l ‡ 2)!
\ufffd \ufffd1=2
r
d
dr
ÿ r
2
‡ n
\ufffd \ufffd
3 r
d
dr
ÿ r
2
‡ nÿ 1
\ufffd \ufffd
\ufffd \ufffd \ufffd r d
dr
ÿ r
2
‡ l ‡ 2
\ufffd \ufffd
r leÿr=2 (6:50)
Just as equation (6.46) can be used to go \u2018up the ladder\u2019 to obtain Sn, l from
170 The hydrogen atom
Snÿ1, l, equation (6.44) allows one to go \u2018down the ladder\u2019 and obtain Snÿ1, l
from Snl. Taking the positive square root in going from equation (6.43) to
(6.44) is consistent with taking the positive square root in going from equation
(6.45) to (6.46); the signs of the functions Snl are maintained in the raising and
lowering operations. In all cases the ladder operators yield normalized eigen-
functions if the starting eigenfunction is normalized.
The radial factors of the hydrogen-like atom total wave functions \u142(r, \u141, j)
are related to the functions Snl(r) by equation (6.23). Thus, we have
R10 ˆ 2 Z
aì
\ufffd \ufffd3=2
eÿr=2
R20 ˆ 1
2

2
p Z
aì
\ufffd \ufffd3=2
(2ÿ r)eÿr=2
R30 ˆ 1
9

3
p Z
aì
\ufffd \ufffd3=2
(6ÿ 6r‡ r2)eÿr=2
..
.
R21 ˆ 1
2

6
p Z
aì
\ufffd \ufffd3=2
reÿr=2
R31 ˆ 1
9

6
p Z
aì
\ufffd \ufffd3=2
(4ÿ r)reÿr=2
R41 ˆ 1
32

15
p Z
aì
\ufffd \ufffd3=2
(20ÿ 10r‡ r2)reÿr=2
..
.
and so forth.
A more extensive listing appears in Table 6.1.
Radial functions in terms of associated Laguerre polynomials
The radial functions Snl(r) and Rnl(r) may be expressed in terms of the
associated Laguerre polynomials L
j
k(r), whose definition and mathematical
properties are discussed in Appendix F. One method for establishing the
relationship between Snl(r) and L
j
k(r) is to relate Snl(r) in equation (6.50) to
the polynomial L
j
k(r) in equation (F.15). That process, however, is long and
tedious. Instead, we show that both quantities are solutions of the same
differential equation.
6.3 The radial equation 171
Table 6.1. Radial functions Rnl for the hydrogen-like atom for
n ˆ 1 to 6. The variable r is given by r ˆ 2Zr=naì
R10 ˆ 2(Z=aì)3=2eÿr=2
R20 ˆ (Z=aì)
3=2
2

2
p (2ÿ r)eÿr=2
R21 ˆ (Z=aì)
3=2
2

6
p reÿr=2
R30 ˆ (Z=aì)
3=2
9

3
p (6ÿ 6r‡ r2)eÿr=2
R31 ˆ (Z=aì)
3=2
9

6
p (4ÿ r)r eÿr=2
R32 ˆ (Z=aì)
3=2
9

30
p r2 eÿr=2
R40 ˆ (Z=aì)
3=2
96
(24ÿ 36r‡ 12r2 ÿ r3)eÿr=2
R41 ˆ (Z=aì)
3=2
32

15
p (20ÿ 10r‡ r2)r eÿr=2
R42 ˆ (Z=aì)
3=2
96

5
p (6ÿ r)r2 eÿr=2
R43 ˆ (Z=aì)
3=2
96

35
p r3 eÿr=2
R50 ˆ (Z=aì)
3=2
300

5
p (120ÿ 240r‡ 120r2 ÿ 20r3 ‡ r4)eÿr=2
R51 ˆ (Z=aì)
3=2
150

30
p (120ÿ 90r‡ 18r2 ÿ r3)r eÿr=2
R52 ˆ (Z=aì)
3=2
150

70
p (42ÿ 14r‡ r2)r2 eÿr=2
R53 ˆ (Z=aì)
3=2
300

70
p (8ÿ r)r3 eÿr=2
R54 ˆ (Z=aì)
3=2
900

70
p r4 eÿr=2
R60 ˆ (Z=aì)
3=2
2160

6
p (720ÿ 1800r‡ 1200r2 ÿ 300r3 ‡ 30r4 ÿ r5)eÿr=2
172 The hydrogen atom
We observe that the solutions Snl(r) of the differential equation (6.24)
contain the factor r leÿr=2. Therefore, we define the function Fnl(r) by
Snl(r) ˆ Fnl(r)r leÿr=2
and substitute this expression into equation (6.24) with º ˆ n to obtain
r
d2 Fnl
dr2
‡ (2l ‡ 2ÿ r) dFnl
dr
‡ (nÿ l ÿ 1)Fnl ˆ 0 (6:51)
where we have also divided the equation by the common factor r.
The differential equation satisfied by the associated Laguerre polynomials is
given by equation (F.16) as
r
d2 L
j
k
dr2
‡ ( j‡ 1ÿ r) dL
j
k
dr
‡ (k ÿ j)L jk ˆ 0
If we let k ˆ n‡ l and j ˆ 2l ‡ 1, then this equation takes the form
r
d2 L2 l‡1n‡ l
dr2
‡ (2l ‡ 2ÿ r) dL
2 l‡1
n‡ l
dr
‡ (nÿ l ÿ 1)L2 l‡1n‡ l ˆ 0 (6:52)
We have already found that the set of functions Snl(r) contains all the
solutions to (6.24). Therefore, a comparison of equations (6.51) and (6.52)
shows that Fnl is proportional to L
2 l‡1
n‡1 . Thus, the function Snl(r) is related to
the polynomial L2 l‡1n‡ l (r) by
Snl(r) ˆ cnlr leÿr=2 L2 l‡1n‡ l (r) (6:53)
The proportionality constants cnl in equation (6.53) are determined by the
normalization condition (6.25). When equation (6.53) is substituted into (6.25),
we have
Table 6.1. (cont.)
R61 ˆ (Z=aì)
3=2
432

210
p (840ÿ 840r‡ 252r2 ÿ 28r3 ‡ r4)r eÿr=2
R62 ˆ (Z=aì)
3=2
864

105
p (336ÿ 168r‡ 24r2 ÿ r3)r2 eÿr=2
R63 ˆ (Z=aì)
3=2
2592

35
p (72ÿ 18r‡ r2)r3 eÿr=2
R64 ˆ (Z=aì)
3=2
12 960

7
p (10ÿ r)r4 eÿr=2
R65 ˆ (Z=aì)
3=2
12 960

77
p r5 eÿr=2
6.3 The radial equation 173
c2nl
…1
0
r2 l‡1eÿr[L2 l‡1n‡ l (r)]
2 dr ˆ 1
The value of the integral is given by equation (F.25) with Æ ˆ n‡ l and
j ˆ 2l ‡ 1, so that
c2nl
2n[(n‡ l)!]3
(nÿ l ÿ 1)! ˆ 1
and Snl(r) in equation (6.53) becomes
Snl(r) ˆ ÿ (nÿ l ÿ 1)!
2n[(n‡ l)!]3
\ufffd \ufffd1=2
r leÿr=2 L2 l‡1n‡ l (r) (6:54)
Taking the negative square root maintains the sign of Snl(r).
Equations (6.39) and (F.22), with Snl(r) and L
j
k(r) related by (6.54), are
identical. From equations (F.23) and (F.24), we find…1
0
Snl(r)Sn\ufffd1, l(r)r2 dr ˆ ÿ12

(nÿ l)(n‡ l ‡ 1)
n(n‡ 1)
s
…1
0
Snl(r)Sn9, l(r)r2 dr ˆ 0, n9 6ˆ n, n\ufffd 1
The normalized radial functions Rnl(r) may be expressed in terms of the
associated Laguerre polynomials by combining equations (6.22), (6.23), and
(6.54)
Rnl(r) ˆ ÿ

4(nÿ l ÿ 1)!Z3
n4[(n‡ l)!]3a3ì
s
2Zr
naì
\ufffd \ufffd l
eÿZr=na0 L2 l‡1n‡ l (2Zr=naì) (6:55)
Solution for positive energies
There are also solutions to the radial differential equation (6.17) for positive
values of the energy E, which correspond to the ionization of the hydrogen-like
atom. In the limit r!1, equations (6.17) and (6.18) for positive E become
d2 R(r)
dr2
‡ 2ìE
"2
R(r) ˆ 0
for which the solution is
R(r) ˆ ce\ufffdi(2ìE)1=2 r="
where c is a constant of integration. This solution has oscillatory behavior at
infinity and leads to an acceptable, well-behaved eigenfunction of equation
(6.17) for all positive eigenvalues E. Thus, the radial equation (6.17) has a
continuous range of positive eigenvalues as well as the discrete set (equation
(6.48)) of negative eigenvalues. The corresponding eigenfunctions represent
174 The hydrogen atom
unbound or scattering states and are useful in the study of electron\u2013ion
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.
Infinite nuclear mass
The energy levels En and the radial functions Rnl(r) depend on the reduced
mass ì of the two-particle system
ì ˆ mN me
mN ‡ me ˆ
me
1‡ me
mN
where mN is the nuclear mass and me is the electronic mass. The value of me is
9:109 39 3 10ÿ31 kg. For hydrogen, the nuclear mass is the protonic mass,
1:672 62 3 10ÿ27 kg, so that ì is 9:1044 3 10ÿ31 kg. For heavier hydrogen-like
atoms, the nuclear mass is, of course, greater than the protonic mass. In the
limit mN !1, the reduced mass and the electronic mass are the same. In the
classical