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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number n, so we may drop
the subscript l. A comparison with equation (6.57) shows that the total energy
is equal to one-half of the average potential energy
En ˆ 12hVin (6:80)
Since the total energy is the sum of the kinetic energy T and the potential
energy V, we also have the expression
Tn ˆ ÿEn ˆ Z
2e92
2a0 n2
(6:81)
The relationship En ˆ ÿTn ˆ (Vn=2) is an example of the quantum-mechani-
cal virial theorem.
6.5 Spectra
The theoretical results for the hydrogen-like atom may be related to experimen-
tally measured spectra. Observed spectral lines arise from transitions of the
atom from one electronic energy level to another. The frequency í of any given
spectral line is given by the Planck relation
í ˆ (E2 ÿ E1)=h
where E1 is the lower energy level and E2 the higher one. In an absorption
spectrum, the atom absorbs a photon of frequency í and undergoes a transition
from a lower to a higher energy level (E1 ! E2). In an emission spectrum, the
process is reversed; the transition is from a higher to a lower energy level
(E2 ! E1) and a photon is emitted. A spectral line is usually expressed as a
wave number ~í, defined as the reciprocal of the wavelength º
~í \ufffd 1
º
ˆ í
c
ˆ jE2 ÿ E1j
hc
(6:82)
The hydrogen-like atomic energy levels are given in equation (6.48). If n1 and
n2 are the principal quantum numbers of the energy levels E1 and E2,
respectively, then the wave number of the spectral line is
6.5 Spectra 187
~í ˆ RZ2 1
n21
ÿ 1
n22
\ufffd \ufffd
, n2 . n1 (6:83)
where the Rydberg constant R is given by
R ˆ ìe9
4
4ð"3c
(6:84)
The value of the Rydberg constant varies from one hydrogen-like atom to
another because the reduced mass ì is a factor. It is not appropriate here to
replace the reduced mass ì by the electronic mass me because the errors
caused by this substitution are larger than the uncertainties in the experimental
data. The measured values of the Rydberg constants for the atoms 1H, 4He‡,
7Li2‡, and 9 Be3‡ are listed in Table 6.3. Following the custom of the field of
spectroscopy, we express the wave numbers in the unit cmÿ1 rather than the SI
unit mÿ1. Also listed in Table 6.3 is the extrapolated value of R for infinite
nuclear mass. The calculated values from equation (6.84) are in agreement
with the experimental values within the known number of significant figures
for the fundamental constants me, e9, and " and the nuclear masses mN . The
measured values of R have more significant figures than any of the quantities
in equation (6.84) except the speed of light c.
The spectrum of hydrogen (Z ˆ 1) is divided into a number of series of
spectral lines, each series having a particular value for n1. As many as six
different series have been observed:
n1 ˆ 1, Lyman series ultraviolet
n1 ˆ 2, Balmer series visible
n1 ˆ 3, Paschen series infrared
n1 ˆ 4, Brackett series infrared
n1 ˆ 5, Pfund series far infrared
n1 ˆ 6, Humphreys series very far infrared
Table 6.3. Rydberg constant for
hydrogen-like atoms
Atom R (cmÿ1)
1H 109 677.58
2H (D) 109 707.42
4He‡ 109 722.26
7Li2‡ 109 728.72
9Be3‡ 109 730.62
1 109 737.31
188 The hydrogen atom
Thus, transitions from the lowest energy level n1 ˆ 1 to the higher energy
levels n2 ˆ 2, 3, 4, . . . give the Lyman series, transitions from n1 ˆ 2 to
n2 ˆ 3, 4, 5, . . . give the Balmer series, and so forth. An energy level diagram
for the hydrogen atom is shown in Figure 6.6. The transitions corresponding to
the spectral lines in the various series are shown as vertical lines between the
energy levels.
n 5 ¥
n 5 6
n 5 5
n 5 4
n 5 3
n 5 2
n 5 1
Pfund
series
Brackett
seriesPaschen
series
Balmer
series
Lyman
series
212
210
28
26
24
22
0
214
E
ne
rg
y 
(e
V
)
Continuum
Figure 6.6 Energy levels for the hydrogen atom.
6.5 Spectra 189
A typical series of spectral lines is shown schematically in Figure 6.7. The
line at the lowest value of the wave number ~í corresponds to the transition
n1 ! (n2 ˆ n1 ‡ 1), the next line to n1 ! (n2 ˆ n1 ‡ 2), and so forth. These
spectral lines are situated closer and closer together as n2 increases and
converge to the series limit, corresponding to n2 ˆ 1. According to equation
(6.83), the series limit is given by
~í ˆ R=n21 (6:85)
Beyond the series limit is a continuous spectrum corresponding to transitions
from the energy level n1 to the continuous range of positive energies for the
atom.
The reduced mass of the hydrogen isotope 2H, known as deuterium, slightly
differs from that of ordinary hydrogen 1H. Accordingly, the Rydberg constants
for hydrogen and for deuterium differ slightly as well. Since naturally occurring
hydrogen contains about 0.02% deuterium, each observed spectral line in
hydrogen is actually a doublet of closely spaced lines, the one for deuterium
much weaker in intensity than the other. This effect of nuclear mass on spectral
lines was used by Urey (1932) to prove the existence of deuterium.
Pseudo-Zeeman effect
The influence of an external magnetic field on the spectrum of an atom is
known as the Zeeman effect. The magnetic field interacts with the magnetic
moments within the atom and causes the atomic spectral lines to split into a
number of closely spaced lines. In addition to a magnetic moment due to its
orbital motion, an electron also possesses a magnetic moment due to an
intrinsic angular momentum called spin. The concept of spin is discussed in
Chapter 7. In the discussion here, we consider only the interaction of the
external magnetic field with the magnetic moment due to the electronic orbital
motion and neglect the effects of electron spin. Thus, the following analysis
¥í~
Figure 6.7 A typical series of spectral lines for a hydrogen-like atom shown in terms of
the wave number ~í.
190 The hydrogen atom
does not give results that correspond to actual observations. For this reason, we
refer to this treatment as the pseudo-Zeeman effect.
When a magnetic field B is applied to a hydrogen-like atom with magnetic
moment M, the resulting potential energy V is given by the classical expression
V ˆ ÿM : B ˆ ìB
"
L : B (6:86)
where equation (5.81) has been introduced. If the z-axis is selected to be
parallel to the vector B, then we have
V ˆ ìB BLz=" (6:87)
If we replace the z-component of the classical angular momentum in equation
(6.87) by its quantum-mechanical operator, then the Hamiltonian operator H^B
for the hydrogen-like atom in a magnetic field B becomes
H^ B ˆ H^ ‡ ìB B
"
L^z (6:88)
where H^ is the Hamiltonian operator (6.14) for the atom in the absence of the
magnetic field. Since the atomic orbitals \u142nlm in equation (6.56) are simultan-
eous eigenfunctions of H^ , L^2, and L^z, they are also eigenfunctions of the
operator H^ B. Accordingly, we have
H^ B\u142nlm ˆ H^ ‡ ìB B
"
L^z
\ufffd \ufffd
\u142nlm ˆ (En ‡ mìB B)\u142nlm (6:89)
where En is given by (6.48) and equation (6.15c) has been used. Thus, the
energy levels of a hydrogen-like atom in an external magnetic field depend on
the quantum numbers n and m and are given by
Enm ˆ ÿ Z
2e92
2aìn2
‡ mìB B, n ˆ 1, 2, . . . ; m ˆ 0, \ufffd1, . . . , \ufffd(nÿ 1)
(6:90)
This dependence on m is the reason why m is called the magnetic quantum
number.
The degenerate energy levels for the hydrogen atom in the absence of an
external magnetic field are split by the magnetic field into a series of closely
spaced levels, some of which are non-degenerate while others are still
degenerate. For example, the energy level E3 for n ˆ 3 is nine-fold degenerate
in the absence of a magnetic field. In the magnetic field, this energy level is
split into five levels: E3 (triply degenerate), E3 ‡ ìB B (doubly degenerate),
E3 ÿ ìB B (doubly degenerate), E3 ‡ 2ìB B (non-degenerate),