Chemistry Reference
In-Depth Information
4.1 Effect of the Approximate Hamiltonians
After the publication of the Dirac equation in 1928, Hartree [
68
] was the first to
analyze the distribution of the charge and current in the Dirac formalism, but his
publication did not contain any graphical representations of the electron density.
The first graphical representation of the four-component Dirac electron density was
then presented by White [
69
,
70
], who investigated the angular distribution of the
charge density for hydrogen-like atoms (highly positively charged one-electron
ions). He presented normalized radial charge densities for different orbitals and
made a few qualitative comments on the difference between the Schr
odinger and
the Dirac electron densities. After these pioneering studies, it took more than three
decades until the first more detailed comparison between relativistic and nonrela-
tivistic charge densities was presented by Burke and Grant [
71
] considering a
hydrogen-like atom with a nuclear charge number
Z
€
80, i.e., Hg
79+
. These
authors compared the radial densities obtained with the Dirac-Coulomb Hamilto-
nian to nonrelativistic ones and drew general conclusions from it. The radial density
D
(
r
) is given by:
¼
r
2
ð
p
0
ð
2
p
'C
y
P
2
Q
2
D
ð
r
Þ¼
sin
#
d
#
d
ðÞC ðÞ¼
ð
r
Þþ
ð
r
Þ;
(58)
0
which is thus equal to the sum of the squares of the radial functions
Q
(
r
) and
P
(
r
)of
the four-component wave function. The most general observation corresponds to a
contraction of the relativistic density profiles toward the nucleus, which is most
pronounced for the core-penetrating
s
- and
p
1/2
-shells. The degree of the contraction
is affected by the absolute value of the relativistic azimuthal quantum number
analog
jj¼
1
2
. For large |
k
| values, the relativistic density profiles resemble
strongly the nonrelativistic ones, whereas the contraction is largest for |
k
|
j
þ
1.
A second observation concerns the nodes of the radial wave function, where the
normalized radial electron density vanishes in the nonrelativistic case, but not in the
relativistic one. The relativistic radial electron density is zero only at
r
¼
0.
A number of publications appeared in the following years which investigated
how relativistic effects affect the electron density [
72
-
76
], i.e., the differences
between the Dirac and the Schr
¼
odinger picture. In 1993, van Lenthe, Baerends,
and Snijders reported the regular approximations [
34
,
56
,
57
,
59
] to the Dirac
Hamiltonian and compared the
r
-weighted square root of the electron density for
different orbitals of an uranium atom. For all of the outer shells, the relativistic
€
contraction is fully recovered and
r
rðr
p
obtained from the ZORA Hamiltonian
can hardly be distinguished from the Dirac results, except at the position of the
minima, where it approaches zero in contrast to the Dirac result, which is always
larger than zero. The only significant difference is observed for the innermost shell,
where the ZORA Hamiltonian is able to recover a large part of the relativistic
contraction, but not all of it. Furthermore, the maximum for the 1
s
1/2
orbitals is too
large compared to the Dirac result. Another study by Autschbach and Schwarz [
77
]
