Chemistry Reference
In-Depth Information
way to calculate it within DFT by expressing it as a function of the derivative of the
total energy with respect to the nuclear radius [
101
,
114
].
In the context of approximate relativistic Hamiltonians, Mastalerz et al. [
66
]
analyzed the DKH electron density at the position of the nucleus for different orders
of the DKH transformation and investigate the effect of picture-change corrections
and different nuclear charge distribution models. We select some results of this
study considering the hydrogen-like Hg
79+
highly charged ion for a more detailed
discussion. Because the study [
66
] focused on the electron density at the position of
the nucleus, the authors decided to analyze the 1/
r
2
weighted radial density [of
(
58
)], which is depicted in Fig.
6
for the Hg
79+
ion. The authors compared the
weighted radial density obtained from fully numerical four-component and scalar-
relativistic DKH(2,0), DKH(5,0), DKH(2,2), and DKH(9,9) calculations. The
curves for point-charge nuclei feature in both cases [Dirac and DKH(n,0)] the
weak singularity at the position of the nucleus, and the position-dependent error
is clearly visible. The DKH(2,0) and DKH(5,0) curves cannot be distinguished
from each other. Hence, already a low-order transformation of the orbitals is
very accurate. In the case of a finite-nucleus model, the weighted radial density
approaches finite values at the position of the nucleus in all cases, which can easily
1e+08
9e+07
DKH(5,0), fn, -3530.397 a.u.
Dirac, fn, -3529.440 a.u.
Dirac, pn, -3532.192 a.u.
DKH(5,0), pn, -3532.367 a.u.
DKH(5,0), pn, lbs, -3532.454 a.u.
DKH(5,0), fn, lbs, -3530.397 a.u.
DKH(2,0), pn, lbs, -3523.321 a.u.
DKH(2,0), fn, lbs, -3521.878 a.u.
basis set artifact
8e+07
point charge nucleus DKH(n,0)
7e+07
finite nucleus DKH(n,0)
6e+07
5e+07
finite nucleus DKH(2,2)
4e+07
finite nucleus DKH(9,9)
3e+07
point charge nucleus Dirac
DKH(n,0)
2e+07
1e+07
finite nucleus Dirac
0
0
0.0002
0.0004
0.0006
0.0008
0.001
r / bohr
h
Fig. 6 1/
r
2
-weighted radial electron densities
r
1
s
1
=
2
ðÞ¼P
1
s
1
=
2
ðÞ=r
2
þ Q
1
s
1
=
2
ðÞ=r
2
Dirac
ð
Þ
and
of Hg
+79
calculated with different nuclear models and basis sets.
The DKH(
n
,0) curves are picture-change affected. The abbreviations “pn” and “fn” denote a point-
charge nucleus and a finite Gaussian charge distribution model, respectively. “lbs” indicates that
the calculation has been performed with a larger basis set that includes more steep functions.
For comparison, the electronic energy in Hartree atomic units is given. Moreover, picture-change-
corrected DKH(2,2) and DKH(9,9) data of the contact density is included. (The figure is reprinted
with permission from [
66
]. Copyright 2010 Elsevier)
P
1
s
ðÞ=r
2
DKH
r
1
s
ðÞ¼
ð
Þ
ðÞ
n
;
0
