Isomer shift
The isomer shift δ is due to the Coulomb interaction between the
absorbing nucleus and the surrounding electrons, expressed as
\(\delta=\alpha\left[\left|\Psi\left(0\right)\right|_{A}^{2}-C\right]\)with\(\alpha=\left(\frac{3Ze^{2}cR^{2}}{5\varepsilon_{0}E_{0}}\right)\frac{\Delta R}{R}\)(2)
showing the dependence of the isomer shift on the electron density of
the absorber nucleus, expressed as the square of the wavefunction\(\Psi\)(0) at the nucleus A multiplied with the elemental chargee which is included in the constant α.71 The
term C describes the density at the source nucleus, which is
approximated as constant
(\(C=\left|\Psi\left(0\right)\right|_{\text{source}}^{2}\)). In
addition to the elemental charge e , the constant α contains the
atomic number Z , the radius of the nucleus R , the nuclear
transition energy E 0, and the electric constantε 0 and the speed of light c as natural
constants. ΔR /R describes the relative change of the
nuclear radius upon excitation, which is determined experimentally or
with the help of quantum chemical calculations of the isomer
shift.71 Overall, the isomer shift thus depends
chiefly on the electronic charge density at the absorber nucleus. While
the electron density is the key quantity in computational Mössbauer
spectroscopy, the interpretation of Mössbauer spectra relies on chemical
concepts that are mostly formulated in terms of molecular
orbitals.14,72
Only electrons in core and valence s-orbitals have a finite probability
density to exist at the position of the nucleus; a direct influence of
the valence molecular orbitals on the isomer shift is therefore only
possible via their s-orbital character. An electron in any orbital with
a higher angular momentum (p, d, f, etc.) has zero probability density
at the position of the nucleus due to these orbitals’ nodal planes.
Regardless, the higher angular momentum valence electrons influence the
isomer shift indirectly due to shielding of the nuclear charge; for
instance, higher iron oxidation states will shield the nuclear charge
less effectively, leading to a higher s-density at the nucleus and thus
a lower isomer shift (see Figure 2A). Note that for57Fe, ΔR /R in Eq. (2) is negative,
leading to an inverse relationship between s-electron density and isomer
shift.
Chemical factors that influence the isomer shift include the oxidation
state of iron, iron-ligand bond lengths, covalency and nature of its
bonds, electronegativity of the ligands and shielding due to the 3d
orbital occupation pattern. All of these factors are important and
should be evaluated carefully for a complete interpretation of isomer
shifts, however it appears futile to attempt to fully disentangle all
individual contributions. A thorough discussion of these factors is
presented in Ref. 14.
To compute isomer shifts with DFT, the central quantity is the electron
density at the point of the nucleus. From a regression analysis of
computed electron densities against experimental isomer shifts, fit
parameters a and b are extracted according to:
\(\delta=a+b\left[\rho\left(0\right)-c\right]\)(3)
The parameter c can be introduced for convenience. With this
approach, correlation lines with R2-values up to
0.9821 and maximum deviations of ca. 0.1 mm
s−1 have been obtained with density functional
theory.21,22,72 The nucleus is approximated as a point
charge; studies using finite nuclei models have not shown any
significant improvement. The question of how the electron density is
obtained14,72 has been detailed in textbooks and
dedicated reviews and therefore we will briefly comment only on two
points central to computational Mössbauer spectroscopy: the choice of
basis set and the choice of relativistic corrections.
Clearly, the basis set needs to be sufficiently large to adequately
describe the contact density, where a cusp in the electron density will
occur that is inherently difficult to describe with Gaussian basis sets.
From previous calibration studies,14,19-22,24 triple-ζ
basis sets such as def2-TZVP have shown good agreement with experiment
at reasonable computational cost. A popular choice for iron is the
CP(PPP) basis set14 that was developed specifically
for the description of core properties. A direct comparison of an
all-element def2-TZVP description vs. CP(PPP) on iron and def2-TZVP on
all other elements shows that standard deviations and
R2 values are overall better with the CP(PPP) basis
set, albeit at a somewhat increased computational
cost.19,22,72 These basis set choices will not
reproduce the absolute electron density at the nuclear position; however
although they do yield significant deviations from the experimental
value these are highly systematic and result in a satisfactory
correlation with experiment.14
Relativistic effects have been shown to vary little for different
electronic configurations allowing the introduction of a constant
scaling factor (not included in Equation 2). This has been discussed
extensively in the literature.14 It was furthermore
shown that there is negligible variation of 1s and 2s electron densities
as compared with the small variation in electron densities assigned to
3s orbitals and the substantial variation of 4s-like electron
densities.14 These data were also used to explain the
success of neglecting scalar relativistic effects, which have little
influence on the description of the valence
orbitals.14
Instead of a calibration of the computed contact density,
Filatov73,74 has presented an approach for the
calculation of isomer shifts from first principles, which is mentioned
only briefly for completeness. Using a finite nucleus model, the isomer
shift is expressed using the derivative of the electronic energy with
respect to the radius of the finite nucleus. Importantly, relativistic
and electron correlation effects are incorporated
immediately.75 This approach was used to determine α
in Eq. 2.76
An important point in the comparison of experimental and computational
isomer shifts is its temperature dependence, which emerges from equation
(1): higher temperatures result in larger motions of the nucleus, hence
larger values for <x 2>
and lower f -factors.1 Note that
<x 2> and thus fcan show anisotropy.71 With increasing temperatures,
the second-order Doppler shift appears due to significant thermal
motions of the source and absorber nuclei which lead to a relativistic
shift in the γ-photon proportional to their mean square
velocity.1 While at temperatures of up to 77 K, this
effect usually contributes less than –0.02 mm s−1,
the influence at room temperature can be on the order of –0.1 mm
s−1, i.e. on par with common deviations between
experiment and DFT prediction. Additionally, low-lying excited states
with modified electronic structures and hence different Mössbauer
parameters may become significantly populated at higher temperatures. A
fair comparison between computational and experimental isomer shifts is
therefore guaranteed only at temperatures of a few
Kelvin.20 Since computational uncertainties will
realistically exceed the influence of the temperature, comparison with
experimental data obtained below 80 K appears reasonable.