Chemistry Reference
In-Depth Information
Figure 8
Singlet energy level diagram for the helium atom. The Rydberg series of
transition frequencies clustered below the ionization threshold can be seen. The
frequencies cluster together, making it difficult to assess the quality of the TDDFT
calculated spectra. As discussed in the text, the quantum defect is preferable for this
purpose.
dotted line. For closed-shell atoms and for any spherical one-electron potential
that decays as
r
at large distances, the bound-state transitions form a
Rydberg series with frequencies
1
=
1
o
nl
¼
I
½
61
2
2
ð
n
m
nl
Þ
where
I
is the ionization potential, and
m
nl
is called the quantum defect (QD).
Quantum defect theory was developed by Ham
238
and Seaton
239
before the
Hohenberg-Kohn theorem.
1
The great value of the quantum defect is its ability to capture all the
information about the entire Rydberg series of transitions in a single slowly
varying function of energy,
, which can often be fit by a straight
line or parabola. We compile extremely accurate singlet and triplet values
from wave function calculations
201
for the helium atom in Table 11,
along with results
156
from the exact ground-state KS potential shown in
Figure 2. For each column, the transition frequencies are on the left, while
the corresponding quantum defects are on the right. Note how small the dif-
ferences between transitions become as one climbs up the ladder, and yet the
quantum defect remains finite and converges to a definite value.
m
l
ð
E
¼
o
I
Þ
