5.27. Mössbauer Parameters

\(^{57}\)Fe Mössbauer spectroscopy probes the transitions of the nucleus between the \(I = \frac{1}{2}\) ground state and the \(I = \frac{3}{2}\) excited state at 14.4 keV above the ground state. The important features of the Mössbauer spectrum are the isomer shift (\(\delta\)) and the quadrupole splitting (\(\Delta E_\mathrm{Q}\)). An idealized spectrum is shown in Fig. 5.64.

Fig. 5.64 An idealized Mössbauer spectrum showing both the isomer shift, \(\delta\), and the quadrupole splitting, \(\Delta E_\mathrm{Q}\).

5.27.1. Isomer Shift

The isomer shift measures the shift in the energy of the \(\gamma\)-ray absorption relative to a standard, usually Fe foil. The isomer shift is sensitive to the electron density at the nucleus, and indirectly probes changes in the bonding of the valence orbitals due to variations in covalency and 3d shielding. Thus, it can be used to probe oxidation and spin states, and the coordination environment of the iron.

Both the isomer shift and quadrupole splitting can be successfully predicted using DFT methods. The isomer shift is directly related to the s electron density at the nucleus and can be calculated using the formula

(5.156)\[\delta = \alpha (\rho_0 - C) + \beta \]

where \(\alpha\) is a constant that depends on the change in the distribution of the nuclear charge upon absorption, and \(\rho_0\) is the electron density at the nucleus [711]. The constants \(\alpha\) and \(\beta\) are usually determined via linear regression analysis of the experimental isomer shifts versus the theoretically calculated electron density for a series of iron compounds with various oxidation and spin states. Since the electron density depends on the functional and basis set employed, fitting must be carried out for each combination used. A compilation of calibration constants (\(\alpha\), \(\beta\) and \(C\)) for various methods was assembled.[712] Usually an accuracy of better than 0.10 mm s\(^{-1}\) can be achieved for DFT with reasonably sized basis sets.

5.27.2. Quadrupole splitting

The quadrupole splitting arises from the interaction of the nuclear quadrupole moment of the excited state with the electric field gradient at the nucleus. The former is related to the non-spherical charge distribution in the excited state. As such it is extremely sensitive to the coordination environment and the geometry of the complex.

The quadrupole splitting is proportional to the largest component of the electric field gradient (EFG) tensor at the nucleus and can be calculated using the formula:

(5.157)\[\Delta E_\mathrm{Q} = \frac{1}{2}eQV_{zz} \left( 1+\frac{\eta^2}{3} \right)^{\frac{1}{2} } \]

where \(e\) is the electrical charge of an electron and \(Q\) is the nuclear quadrupole moment of Fe (approximately 0.16 barns). \(V_{xx}\), \(V_{yy}\) and \(V_{zz}\) are the electric field gradient tensors and \(\eta\), defined as

(5.158)\[\eta = \left| \frac{V_{xx} - V_{yy} }{V_{zz} } \right|\]

is the asymmetry parameter in a coordinate system chosen such that \(|V_{zz}| \geq |V_{yy}| \geq |V_{xx}|\).

The electric field gradient tensor is closely related to the dipole contribution to the hyperfine coupling. The main differences are that the electron instead of the spin density enters its calculation and that it contains a nuclear contribution due to the surrounding nuclei. It is calculated from

(5.159)\[\begin{split}\begin{array}{l} V_{\mu \nu } \left( N \right)=-\sum\limits_{kl} { P_{kl} } \left\langle {\phi_{k} \left|{ r_{N}^{-5} \left({ 3\vec{{r} }_{N\mu } \vec{{r} }_{N\nu } -\delta_{\mu \nu } r_{N}^{2} } \right)} \right|\phi_{l} } \right\rangle\\ \hspace{1.5cm} +\sum\limits_{A\ne N} { Z_{A} \vec{{R} }_{AN}^{-5} \left({ 3\vec{{R} }_{AN\mu } \vec{{R} }_{AN\nu } -\delta_{\mu \nu } R_{AN}^{2} } \right)} \\ \end{array} \end{split}\]

with \(Z_{A}\) as the nuclear charge of nucleus \(A\) and \(\vec{{R} }_{AN}\) as a vector of magnitude \(R_{AN}\) that points from nucleus \(A\) to nucleus \(N\). \(\mathrm{\mathbf{P} }\) is the first order density matrix.

5.27.3. Basic usage

An example of how to calculate the electron density and quadrupole splitting of an iron center is as follows:

%eprnmr
  nuclei = all Fe {fgrad, rho}
end

If a basis set with tight functions such as the CP(PPP) is employed, one might have to increase the radial integration accuracy for the iron atom. From ORCA 5.0, this is considered during grid construction and the defaults should work very well. However for very problematic cases it can be increased by controlling the SPECIALGRIDINTACC flag under %METHOD (see Sec. The Radial Grid Scheme for details).

The output file should contain the following lines, where you obtain the calculated quadrupole splitting directly and the RHO(0) value (the electron density at the iron nucleus). To obtain the isomer shift one has to insert the RHO(0) value into the appropriate linear equation (Eq. (5.156)).

Moessbauer quadrupole splitting parameter (proper coordinate system)
e**2qQ  = -0.406 MHz = -0.035 mm/s
eta     = 0.871
Delta-EQ=(1/2{e**2qQ}*sqrt(1+1/3*eta**2) = -0.227 MHz = -0.020 mm/s
RHO(0)=  11581.352209571 a.u.**-3   # the electron density at the Fe nucleus.

Note

• Following the same procedure, Mössbauer parameters can be computed with the CASSCF wavefunction. In case of a state-averaged CASSCF calculation, the averaged density is used in the subsequent Mössbauer calculation.

5.27.4. Keywords

All properties tied to a given nucleus in the molecule (NMR shielding, hyperfine coupling, quadrupolar coupling) share a common input syntax. This syntax consists of a selection (the nuclei for which you want to do something) and a list of actions you want the program to take for these nuclei inside curly brackets. A complete list of possible keywords for the eprnmr module can be found in EPRNMR - keywords for magnetic properties. It’s best explained on an example. The following

%eprnmr
  nuclei = all O { aiso, adip, fgrad, shift, ist=17 }
end

would select all oxygen atoms in the molecule, pick the \(^{17}O\) isotope, and request isotropic + spin-dipolar hyperfine couplings, the electric field gradient, and NMR shielding to be calculated for them.

Warning

All nuclei specified on one line will be assigned with the same isotopic mass (and other parameters)!

Table 5.18 %eprnmr block input keywords relevant for Mössbauer calculations

Keyword	Selection	What to do	Description
nuclei	all <type>		Selects all nuclei of element <type>
	i1,i2,i3...		Selects atoms i1, i2, i3… (starts at 1!)
		{ rho   }	Calculates density at nucleus for selection
		{ fgrad }	Calculates EFG for selection
		{ ist=<N> }	Selects isotope <N> for selection
		{ QQQ=<N> }	Sets quadrupole moment to <N> for selection
		{ III=<N> }	Sets nuclear spin to <N> for selection
		{ fgrad, rho, ist=57 }	Example requesting multiple actions for selection