Chemistry Reference
In-Depth Information
the vibrational energy depends on the second derivative [
99
,
100
]. The contact
density plays also an important role in M
ossbauer spectroscopy, because it can be
related to the chemical isomer shift [
73
,
74
,
101
], which was discovered in 1960
[
102
]. It depends on the differences of the contact density of the emitter source and
the probe [
103
]. A more detailed description of the basic principles of M
€
€
ossbauer
spectroscopy and contact densities can be found in [
104
]. In order to be able to
calculate these properties for atoms of the whole periodic table, one must ensure
that the contact density can be calculated with sufficient accuracy.
The contact density is dominated by the core-penetrating
s
-orbitals that are
strongly affected by relativistic effects. An adequate description requires therefore
a fully relativistic treatment of the electrons. In addition, the choice of the nuclear
charge distribution model has an effect on the accuracy of the calculated electron
density. In the nonrelativistic picture, the nuclei are considered to be point charges,
because it can be shown that the different nuclear charge distribution models
usually yield negligible energy differences when compared to point-like nuclei -
even for heavy nuclei [
105
]. The electron density features in this case a cusp at the
location of the nucleus which is described by Kato's cusp condition [
106
]. In a fully
relativistic description, the choice of the nuclear charge distribution model becomes
important. For heavy atoms, the point-like description of the nuclei causes signifi-
cant errors, whereas the differences between the available nuclear charge distribu-
tion models are small. For a comprehensive review on the available models, we
recommend [
107
]. There are in general two possibilities to include these models
into calculations. Either the calculation is performed for point-like nuclei and
effects of finite nuclei are considered via perturbation theory, or they are directly
incorporated in the calculation. The perturbation theory approach can be found in
early studies [
97
,
98
] on contact densities, whereas the direct consideration should
be more consistent [
108
,
109
]. For the major part of the practical applications, it is
sufficient to use a simple model for the positive charge distribution in the atomic
nucleus [
105
,
110
]. Two simple models are the homogeneous charge distribution
and the Gaussian charge distribution [
108
]. Also the three parameter Fermi model
[
111
] has been used.
The first attempts to obtain fully relativistic electron density distributions con-
sidering finite-nucleus effects can be dated back to Rosenthal et al. [
97
] and Breit
[
98
], who included the effects of a finite nuclear charge distribution via perturbation
theory. A direct treatment of the finite nuclear charge distribution model using a
three parameter Fermi model was then first presented by Fricke and Waber [
108
],
who calculated M
ossbauer isomer shifts.
In the case of solids, properties are calculated within solid-state density functional
theory, where mostly the simple model of an uniformly charged sphere is used, e.g.,
as shown by Svane and Antoncik [
112
]. A recent study by Mastalerz et al. [
109
]
investigated the basis set convergence for the calculation of contact densities at
finite nuclei. A most recent fully relativistic investigation of correlation effects on
the contact density with CCSD(T) was presented for mercury compounds in [
113
].
As a final remark to the calculation of the isomer shift, we refer to an alternative
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