Atomic Spectra
§4.1 Spectra of hydrogenic atoms
Each element emits light of specific energies when excited by an electric discharge or
heat. Transitions observed in the emission spectrum of atomic hydrogen are shown as
follows:
(Miessler, et al., “Inorganic Chemistry”, Figure2.2)
When an electric discharge is passed through gaseous hydrogen, the H2 molecules are
dissociated and the energetically excited H atoms that are produced emit light of
discrete frequencies, producing a spectrum of a series of lines.
* Balmer series: red at 656.3 nm, blue-green at 486.1 nm, blue violet at 434.1 nm and
violet at 410.2 nm.
* The longest wavelength line in the Lyman series has λ = 121.57 nm or 1/λ = 82.258
cm-1.
(Atkins’ Physical Chemistry, 9e, Fig9.1)
The energies of light emitted by the hydrogen atom could be described by the
equation,
 1
1 
E  RH  2  2 
 nl nh 
where RH is the Rydberg constant for hydrogen (= 2.179 × 10-18 J = 13.61 eV =
109677 cm-1); nl  nh , are integers (l for lower lever and h for higher lever).
* The inverse of emission is absorption. An infinite quantum number (n  ∞)
indicates a zero-energy state of infinitely separated, stationary electron and nucleus,
meaning that the electron is no longer part of the atom, and the electron and the
nucleus are separate entities. Beginning from the ground state or other lower states
and ending at the state with n  ∞ corresponds to ionization of the hydrogen atom,
E  RH .
The energy emitted by hydrogenic atoms (or hydrogen-like atoms, the exotic
one-electron situation such as He+, Li+2, Be+3) can be found from the equation:
 1
2 2 Z 2 e 4
1 
depends on the
E  R 2  2  , where the Rydberg constant R 
(40 ) 2 h 2
 nl nh 
reduced mass of the electron/nucleus combination, and 4 0 is the permittivity of a
vacuum, ( 4 0 )-1 = 8.98755 × 109 J m/C2.
The Ritz combination principle states that the wave number of any spectral line (of
any atom, not just hydrogenic atoms) is the difference between two terms. The Ritz
combination principle applies to all types of atoms and molecules, but only for
hydrogenic atoms do the terms have the simple form－constant/n2.
* The energy of spectral lines can be measured with high precision; as an example,
recent emission spectral data of hydrogen atoms in the solar corona indicated a
difference between nh  2 and nl  1 of 82258.9543992821(23) cm-1
(http://physics.nist.gov/asd [2012, August 21], National Institute of Standards and
Technology, Gaithersburg, MD).
例 1. 計算氫原子的電子從能階 nh  3 到 nl  2 放出電磁波的能量 (J)、波長
(nm)、波數 (cm-1)；這道電磁波屬於可見光的紅光。
例 2. Calculate the shortest wavelength line in the Paschen series, and the shortest and
longest wavelength lines in the Lyman series.
例 3. A series of lines in the spectrum of atomic hydrogen lies at 656.46 nm, 486.27
nm, 434.17 nm, and 410.29 nm. What is the wavelength of the next line of the series?
What is the shortest wavelength line in this series?
例 4. The emission spectrum of atomic hydrogen shows lines at 82259, 97492, 102824,
105292, 106632, and 107440cm-1, which correspond to transition to the same lower
state. Determine the ionization energy of the lower state.
§4.2 Selection rules for spectroscopic transitions
When the electron of a hydrogenic atom undergoes a transition from an orbital with
quantum number n1 to another (lower energy) orbital with n2, it discards the excess
energy as a photon with a frequency given by the Bohr frequency condition:

E 2  E1
h
However, since a photon has an intrinsic spin angular momentum corresponding to s
= 1, and the total angular momentum is conserved during the spectroscopic transition,
the change in angular momentum of the electron must compensate for the angular
momentum carried away by the photon. Thus, an electron in a d orbital (l = 2) cannot
make a transition to an s orbital (l = 0) because one photon cannot carry away enough
angular momentum.
A selection rule is a statement about which transitions are allowed, meaning they can
occur, whereas others are forbidden. The selection rules for hydrogenic atoms are
l  1 ; ml  0,  1.
例 1. To what orbitals may a 4s electron make (electric-dipole) allowed radiative
transition?
 to np orbitals only
例 2. Identify the orbitals to which a 4d electron may make (electric-dipole) allowed
radiative transition?
 to any np orbital and to any nf orbital
All atomic species have line spectra, which can be used as a “fingerprint” to identify
the element. However, no atom other than hydrogen has a simple relation for its
spectral frequencies.
Many-electron atoms
Any state of the atom, and any spectra transition, can be specified by using term
symbols. For example, the transition giving rise to the yellow sodium doublet are
3p1
P3/2  3s1
2
2
S1/2,
3p1
P1/2  3s1
2
2
S1/2
(Atkins’ Physical Chemistry, 9e, Fig9.31)
By convention, the upper term proceeds the lower; the corresponding adsorptions are
denoted
2
P3/2  2S1/2,
P1/2  2S1/2
2
(omitting the configurations).
Selection rules which arising from the conservation of angular momentum can be
expressed in terms of term symbols:
ΔS = 0; ΔL = 0, ±1, Δl = ±1; ΔJ = 0, ±1;
but J = 0 ,× J = 0 where “×” denotes a forbidden transition.
ΔS = 0 stems from the fact that the photon, which has a spin of 1, does not affect the
spin directly. The rules about ΔL and Δl express the fact that the orbital angular
momentum of an individual electron must change, but whether or not this results in an
overall change of orbital momentum depends on the coupling.
【The selection rules given above apply when Russell-Saunders coupling scheme is
valid in light atoms, those of lower Z.】