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.