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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two-particle problem of Section 6.1, this limit corresponds to the
nucleus remaining at a fixed point in space.
In most applications, the reduced mass is sufficiently close in value to the
electronic mass me that it is customary to replace ì in the expressions for the
energy levels and wave functions by me. The parameter aì ˆ "2=ìe92 is
thereby replaced by a0 ˆ "2=mee92. The quantity a0 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 me and aì by a0 in
the remainder of this book.
6.4 Atomic orbitals
We have shown that the simultaneous eigenfunctions \u142(r, \u141, j) of the opera-
tors H^ , L^2, and L^z have the form
\u142nlm(r, \u141, j) ˆ jnlmi ˆ Rnl(r)Ylm(\u141, j) (6:56)
where for convenience we have introduced the Dirac notation. The radial
functions Rnl(r) and the spherical harmonics Ylm(\u141, 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
6.4 Atomic orbitals 175
magnetic quantum number m determines the z-component of the angular
momentum. We have found that the allowed values of n, l, and m are
m ˆ 0, \ufffd1, \ufffd2, . . .
l ˆ jmj, jmj ‡ 1, jmj ‡ 2, . . .
n ˆ l ‡ 1, l ‡ 2, l ‡ 3, . . .
This set of relationships may be inverted to give
n ˆ 1, 2, 3, . . .
l ˆ 0, 1, 2, . . . , nÿ 1
m ˆ ÿl, ÿl ‡ 1, . . . , ÿ1, 0, 1, . . . , l ÿ 1, l
These eigenfunctions form an orthonormal set, so that
hn9l9m9jnlmi ˆ änn9ä ll9ämm9
The energy levels of the hydrogen-like atom depend only on the principal
quantum number n and are given by equation (6.48), with aì replaced by a0, as
En ˆ ÿ Z
2e92
2a0 n2
, n ˆ 1, 2, 3, . . . (6:57)
To find the degeneracy gn of En, we note that for a specific value of n there are
n different values of l. For each value of l, there are (2l ‡ 1) different values of
m, giving (2l ‡ 1) eigenfunctions. Thus, the number of wave functions corre-
sponding to n is given by
gn ˆ
Xnÿ1
lˆ0
(2l ‡ 1) ˆ 2
Xnÿ1
lˆ0
l ‡
Xnÿ1
lˆ0
1
The first summation on the right-hand side is the sum of integers from 0 to
(nÿ 1) and is equal to n(nÿ 1)=2 (n terms multiplied by the average value of
each term). The second summation on the right-hand side has n terms, each
equal to unity. Thus, we obtain
gn ˆ n(nÿ 1)‡ n ˆ n2
showing that each energy level is n2-fold degenerate. The ground-state energy
level E1 is non-degenerate.
The wave functions jnlmi for the hydrogen-like atom are often called atomic
orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7, . . . of the
azimuthal quantum number l by the letters s, p, d, f, g, h, i, k, . . . , respectively.
Thus, the ground-state wave function j100i is called the 1s atomic orbital,
j200i is called the 2s orbital, j210i, j211i, and |21 ÿ1l are called 2p orbitals,
and so forth. The first four letters, standing for sharp, principal, diffuse, and
176 The hydrogen atom
fundamental, originate from an outdated description of spectral lines. The
letters which follow are in alphabetical order with j omitted.
s orbitals
The 1s atomic orbital j1si is
j1si ˆ j100i ˆ R10(r)Y00(\u141, j) ˆ 1
ð1=2
Z
a0
\ufffd \ufffd3=2
eÿZr=a0 (6:58)
where R10(r) and Y00(0, j) are obtained from Tables 6.1 and 5.1. Likewise, the
orbital j2si is
j2si ˆ j200i ˆ (Z=a0)
3=2
4

2ð
p 2ÿ Zr
a0
\ufffd \ufffd
eÿZr=2a0 (6:59)
and so forth for higher values of the quantum number n. The expressions for
jnsi for n ˆ 1, 2, and 3 are listed in Table 6.2.
All the s orbitals have the spherical harmonic Y00(\u141, j) as a factor. This
spherical harmonic is independent of the angles \u141 and j, having a value
(2

ð
p
)ÿ1. Thus, the s orbitals depend only on the radial variable r and are
spherically symmetric about the origin. Likewise, the electronic probability
density j\u142j2 is spherically symmetric for s orbitals.
p orbitals
The wave functions for n ˆ 2, l ˆ 1 obtained from equation (6.56) are as
follows:
j2p0i ˆ j210i ˆ (Z=a0)
5=2
4

2ð
p reÿZr=2a0 cos \u141 (6:60a)
j2p1i ˆ j211i ˆ 1
8ð1=2
Z
a0
\ufffd \ufffd5=2
reÿZr=2a0 sin\u141 eij (6:60b)
j2pÿ1i ˆ j21ÿ1i ˆ 1
8ð1=2
Z
a0
\ufffd \ufffd5=2
reÿZr=2a0 sin \u141 eÿij (6:60c)
The 2s and 2p0 orbitals are real, but the 2p1 and 2pÿ1 orbitals are complex.
Since the four orbitals have the same eigenvalue E2, any linear combination of
them also satisfies the Schro¨dinger equation (6.12) with eigenvalue E2. Thus,
we may replace the two complex orbitals by the following linear combinations
to obtain two new real orbitals
6.4 Atomic orbitals 177
Table 6.2. Real wave functions for the hydrogen-like atom. The parameter aì
has been replaced by a0
State Wave function
Spherical coordinates Cartesian coordinates
1s
(Z=a0)
3=2
ð
p eÿZr=a0
2s
(Z=a0)
3=2
4

2ð
p 2ÿ Zr
a0
\ufffd \ufffd
eÿZr=2a0
2pz
(Z=a0)
5=2
4

2ð
p reÿZr=2a0 cos \u141 (Z=a0)
5=2
4

2ð
p zeÿZr=2a0
2px
(Z=a0)
5=2
4

2ð
p reÿZr=2a0 sin \u141 cosj (Z=a0)
5=2
4

2ð
p xeÿZr=2a0
2p y
(Z=a0)
5=2
4

2ð
p reÿZr=2a0 sin \u141 sinj (Z=a0)
5=2
4

2ð
p yeÿZr=2a0
3s
(Z=a0)
3=2
81

3ð
p 27ÿ 18 Zr
a0
‡ 2 Z
2 r2
a20
 !
eÿZr=3a0
3pz
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
reÿZr=3a0 cos \u141
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
zeÿZr=3a0
3px
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
reÿZr=3a0 sin \u141 cosj
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
xeÿZr=3a0
3p y
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
reÿZr=3a0 sin \u141 sinj
2(Z=a0)
5=2
81

2ð
p 6ÿ Zr
a0
\ufffd \ufffd
yeÿZr=3a0
3dz2
(Z=a0)
7=2
81

6ð
p r2eÿZr=3a0 (3 cos2\u141ÿ 1) (Z=a0)
7=2
81

6ð
p (3z2 ÿ r2)eÿZr=3a0
3dxz
2(Z=a0)
7=2
81

2ð
p r2eÿZr=3a0 sin \u141 cos \u141 cosj 2(Z=a0)
7=2
81

2ð
p xzeÿZr=3a0
3d yz
2(Z=a0)
7=2
81

2ð
p r2eÿZr=3a0 sin \u141 cos \u141 sinj 2(Z=a0)
7=2
81

2ð
p yzeÿZr=3a0
3dx2ÿ y2
(Z=a0)
7=2
81

2ð
p r2eÿZr=3a0 sin2\u141 cos 2j (Z=a0)
7=2
81

2ð
p (x2 ÿ y2)eÿZr=3a0
3dxy
(Z=a0)
7=2
81

2ð
p r2eÿZr=3a0 sin2\u141 sin 2j 2(Z=a0)
7=2
81

2ð
p xyeÿZr=3a0
178 The hydrogen atom
j2pxi \ufffd 2ÿ1=2(j2p1i ‡ j2pÿ1i) ˆ 1
4(2ð)1=2
Z
a0
\ufffd \ufffd5=2
reÿZr=2a0 sin\u141 cosj (6:61a)
j2p yi \ufffd ÿi2ÿ1=2(j2 p1i ÿ j2pÿ1i) ˆ 1
4(2ð)1=2
Z
a0
\ufffd \ufffd5=2
reÿZr=2a0 sin \u141 sinj
(6:61b)
where equations (A.32) and (A.33) have been used. These new orbitals j2pxi
and j2p yi are orthogonal to each other and to all the other eigenfunctions
jnlmi. The factor 2ÿ1=2 ensures that they are normalized as well. Although
these new orbitals are simultaneous eigenfunctions of the Hamiltonian operator
H^ and of the operator L^2, they are not eigenfunctions of the operator L^z.
If we now substitute equations (5.29a), (5.29b), and (5.29c) into (6.61a),
(6.61b), and (6.60a), respectively, we obtain for the set of three real 2p orbitals
j2pxi ˆ 1
4(2ð)1=2
Z
a0
\ufffd \ufffd5=2
xeÿZr=2a0 (6:62a)
j2p yi ˆ 1
4(2ð)1=2
Z
a0
\ufffd \ufffd5=2
yeÿZr=2a0 (6:62b)
j2pzi ˆ 1
ð1=2
Z
2a0
\ufffd \ufffd5=2
zeÿZr=2a0 (6:62c)
The subscript x, y, or z on a 2p orbital indicates that the angular part of the
orbital has its maximum value along that axis. Graphs of the square of the
angular part of these three functions are presented in Figure 6.2. The mathema-
tical expressions for the real 2p and 3p atomic