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compares the relativistic change of the radial density for hydrogen-like atoms with
a nuclear charge number of
Z
100. These authors compare the Dirac
density to two different approximations, namely the Pauli (which is not discussed
here as it is only of historical importance) and the ZORA Hamiltonians arriving at a
similar conclusion as van Lenthe, Baerends, and Snijders.
DKH electron densities were presented by van W
¼
1 and
Z
¼
€
ullen and Michauk [
78
] who
emphasize in a brief discussion that the electron density obtained from the two-
component wave functions (neglecting picture-change effects) is not equal to the
Dirac density due to the picture-change error. As explained above, the DKH
electron density, approximated by the sum of the squared DKH orbitals, is suffi-
ciently accurate for the valence region of an atom, but the error increases, the
smaller the distance from the nucleus is (this subject is further analyzed in Sect.
5
).
Picture-change-affected DKH density were also analyzed in [
65
], considering a
hydrogen-like mercury atom with
Z
80.
Eickerling et al. [
64
] presented the first systematic investigation of the effects
of an approximate two-component Hamiltonian and the scalar-relativistic DKH10
Hamiltonian on the electron density and its topology by comparing to the four-
component Dirac Hamiltonian and to the nonrelativistic limit, namely the
Schr
¼
odinger Hamiltonian. The study features a comparison of difference electron
densities obtained from three relativistic and the nonrelativistic Hamiltonian for a
homologous series of acetylene complexes M-C
2
H
2
with M
€
Ni,Pd,Pt and in
addition an analysis of the negative Laplacian at the bond critical points (BCPs),
which are minima of the electron density on the bonding axis and maxima on the
axis perpendicular to the bonding axis. The most significant difference between the
four-component Dirac density and the nonrelativistic one, considering all BCPs, is
observed for the M-C
2
H
2
BCP in the case of M
¼
Pt, where it amounts to
0.06 e
˚
3
. The study concludes that scalar-relativistic methods cover most relativ-
istic effects, though there are still differences to the electron densities obtained from
two-component methods. Concerning the negative Laplacians, the deviations can
be larger which makes it a more sensitive measure than the electron density itself.
Furthermore, the size of the relativistic effects is estimated to be of almost the same
size as correlation effects in four-component DFT calculations.
In order to provide a closer look at the accuracy of approximate relativistic
Hamiltonians, we discuss the results for the homologous acetylene complexes given
by Eickerling et al. [
64
] in more detail. The difference electron densities
r
rel.
(
¼
r
)
r
nonrel.
(
) obtained from four-component Dirac, ZORA-SO (including spin-orbit
effects), and scalar-relativistic DKH10 calculations are shown in Fig.
1a, d, g
.
Although the relativistic effects are expected to be most pronounced for the case
of M
r
Pt, one can observe significant differences even for the nickel complex.
The difference electron density map for the nickel complex contains different
circular minima and maxima due to the changes in the radial extension of the
atomic sub-shells. The innermost circular region of positive difference exhibits four
maxima around the nickel atom, from which the one oriented in the direction of the
acetylene ligand is more pronounced in the case of DKH10 than for the ZORA-SO
or the four-component Dirac density. These maxima correspond to local charge
¼
