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First principles theoretical modeling of the isomer shift of Mossbauer spectraKurian, Reshmi

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First principles theoretical modeling

of the isomer shift of Mossbauer

spectra

Reshmi Kurian

Zernike Institute PhD thesis series 2011-01

ISSN 1570-1530

The work described in this thesis has been carried out in the Theo-

retical Chemistry Group- Zernike Institute For Advanced Materials,

University of Groningen, The Netherlands.

Cover design : Charly John

Printed by : Ipskamp Drukkers, Enschede.

Proefschrift Rijksuniversiteit Groningen.

c© R. Kurian, 2011.

RIJKSUNIVERSITEIT GRONINGEN

First principles theoretical modeling

of the isomer shift of Mossbauer

spectra

Proefschrift

ter verkrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus, dr. F. Zwarts,

in het openbaar te verdedigen op

vrijdag 14 januari 2011

om 16.15 uur

door

Reshmi Kurian

geboren op 10 mei 1983

te Thankamany, India

Promotor: Prof. dr. M. Filatov

Beoordelingscommissie: Prof. dr. F. Neese

Prof. dr. L. Visscher

Prof. dr. Y. Garcia

ISBN : 978-90-367-4664-9 Printed version

ISBN : 978-90-367-4666-3 Electronic version

For my family, who offered me unconditional love and support throughout

the course of this thesis

Contents

1 Introduction and Objective 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Isomer Shift . . . . . . . . . . . . . . . . . . . . . 10

1.1.2 Quadrupole splitting . . . . . . . . . . . . . . . . 12

1.1.3 Magnetic Hyperfine Splitting . . . . . . . . . . . 14

1.2 Applications of Mossbauer Spectroscopy . . . . . . . . 15

1.3 Significance of the present work . . . . . . . . . . . . . . 19

1.4 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . 21

2 Theoretical Framework 25

2.1 General quantum mechanical formalism . . . . . . . . . 25

2.1.1 Schrodinger equation . . . . . . . . . . . . . . . 25

2.1.2 Dirac Equation and importance of relativistic ef-

fects . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.3 Hartree Fock theory . . . . . . . . . . . . . . . . 32

2.1.4 Electron correlation treatment . . . . . . . . . . . 34

vii

Contents

2.2 Theory of Mossbauer Isomer Shifts . . . . . . . . . . . . 42

2.2.1 Perturbative treatment of Mossbauer Isomer Shifts 43

2.2.2 Mossbauer Isomer shift as energy derivative . . 45

3 Testing DFT methods in the calculation of MIS 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Computational Details . . . . . . . . . . . . . . . . . . . 54

3.3 Results and discussion . . . . . . . . . . . . . . . . . . . 56

3.3.1 Influence of basis sets on isomer shift values . . 56

3.3.2 Isomer shift variation with different theoretical

levels . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Calibration of 119Sn isomer shift 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 71


4.3 Results and discussion . . . . . . . . . . . . . . . . . . . 78

4.3.1 Calibration of the 119Sn isomer shift . . . . . . . 80

4.3.2 Isomer shift variation in CaSnO3 perovskite . . 85

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Calibration of 57Fe isomer shift 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 91


5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . 96

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Study of the presence of different iron sites in RbMn[Fe(CN)6]·H2O103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 103


viii

Contents 1


6.3.1 Influence of the different oxidation states and

coordination sites . . . . . . . . . . . . . . . . . . 107

6.3.2 Influence of the bond length variations . . . . . 110

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Theoretical investigation of hyperfine parameters of iron-based

superconductors 113

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2 Details of calculations . . . . . . . . . . . . . . . . . . . 116


7.3.1 MIS calculations . . . . . . . . . . . . . . . . . . 120

7.3.2 HFC calculations . . . . . . . . . . . . . . . . . . 124

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8 Conclusions and Outlook 133

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography 141

Summary 152

Samenvatting 154

List of Publications 157

Acknowledgements 158

Chapter 1

Introduction and Objective

Synopsis

The basic ideas behind Mossbauer Spectroscopic technique is outlined in the

present chapter. The recoilless emission and absorption of γ-rays, which is

the key feature of this spectroscopy is explained with illustrations. The un-

derstanding of the electronic structure of the compounds under study, from

the Mossbauer spectra parameters are discussed in detail. The chapter also

includes the objective and scope of this dissertation.

1.1 Introduction

Fifty years ago Rudolf L. Mossbauer discovered the recoilless nu-

clear resonance absorption of γ-rays while working on his doc-

toral thesis. This phenomenon, which rapidly developed into a new

spectroscopic technique is known as Mossbauer effect [1–4]. Over

the last couple of decades, Mossbauer spectroscopy has become one

of the most captivating tools in chemical physics providing informa-

tion about the chemical environment of the resonating nucleus on an

atomic scale [1–4]. The most well-known application is the determi-

nation of iron 57Fe in crystalline and in disordered solid samples. Be-

sides iron, there are many elements in the periodic table which have

4 1. Introduction and Objective

Mossbauer active nuclei [5–8]. The Mossbauer effect has been ob-

served for the elements which are dotted in the periodic table shown

(Figure 1.1).

Figure 1.1: The periodic table showing the Mossbauer active nuclei, thedotted elements.

The phenomenon of recoilless resonance absorption/emission of

γ-rays by nuclei is the basic characteristic of Mossbauer Spectoscopy

[5, 9]. When a nucleus in an excited state of energy Ee undergoes a

transition to the ground state with energy Eg by emitting a γ quan-

tum, E0 = Ee - Eg, the γ quantum energy E0 may be totally absorbed

by a nucleus of the same kind in its ground state. This phenomenon is

called nuclear resonance absorption of γ-rays and is shown schemati-

cally in Figure 1.2. The resonance absorption is observable only if the

emission and absorption lines overlap sufficiently. The mean lifetime

of the excited state τN , determines the width of the resonance lines

(Γ) according to time-energy uncertainity relation, Γ.τN ≥ ~. There-

1.1. Introduction 5

fore, longer lifetimes produce too narrow transition lines and shorter

lifetimes produce too broad transition lines, and the resonance over-

lap between the emission and absorber lines is not possible. The suit-

able lifetimes of the excited nuclear states for the resonance absorption

range from 10−6 to 10−11s. The spectral line shape may be described

by the Lorentzian or Breit-Wigner form [5,9]. For the 14.4 keV level of57Fe, the natural line width Γ is determined using the mean life time τ

= t1/2ln2

= 1.43×10−7 s , which is Γ = ~/2π = 4.55 × 10−9eV.

Figure 1.2: Schematic representation of nuclear resonance absorption of γ-rays.

When a γ-ray is emitted from an excited nucleus of mass M, there

occurs a recoil of the nucleus according to which the nucleus moves

with velocity v in the opposite direction of γ-ray emission. According

to the conservation of momentum, the linear momenta of the nucleus

and the γ quantum are equal and the energy of the emitted γ quantum

are reduced as, Eγ = E0 - ER. The recoil energy can be written as,

ER =p2

2M=

E2γ

2Mc2(1.1)

The range of transition energies Eγ for the occurrence of resonance

absorption is 5-180 keV. However, the difficulties of high-energy nu-

clear transitions can get around by the use of synchrotron radiation


sources [10]. For the Mossbauer transition of 57Fe from the excited to

the ground state (E0 = Ee - Eg = 14.4 keV), the recoil energy ER is eval-

uated to be 1.95 × 10−3 eV according to Eqn. 1.1. This value is about

six orders of magnitude larger than the natural width of the spectral

transition under consideration (Γ = 4.55 × 10−9 eV). Figure 1.3 shows

intensity I(E) as a function of the transition energy for emission and

absorption of γ transition. The recoil effect reduces the transition en-

ergy by ER for the emission process and increases the transition en-

ergy by ER for the absorption process. Therefore, the transition lines

for the emission and absorption are separated by a distance 2ER on the

energy scale as shown in Figure 1.3, which is about 10−6 times larger

than the natural line width Γ. Hence, the overlap between two transi-

tion lines and nuclear resonance absorption is not possible in isolated

atoms or molecules in the gaseous or liquid state.

If the γ-ray emission takes place while both the emitter and absorber

Figure 1.3: The transition lines for emission and absorption in an isolatednuclei, which is separated by 2ER.

nucleus are moving relative to each other, then the γ-photon of energy

Eγ receives a Doppler energy ED, ie, Eγ = E0 - ER + ED. For 57Fe, mov-

1.1. Introduction 7

ing the source at a velocity of 1mm/s towards the sample increases the

energy of the emitted photons by about ten natural line widths. Thus,

the Doppler shift of the emission and absorption lines allows for the

fine tuning of the resonances in Mossbauer experiments [5,9]. The rel-

ativistic Doppler formula for an emitter and absorber with a relative

velocity of υ is,

v = v0

√

1− υ/c

1 + υ/c(1.2)

where c is the speed of light. Expanding Eqn 1.2 into Taylor series

gives the velocity shift, i.e.,

v = v0

(

1− υ/c+1

2υ2/c2 − .......

)

(1.3)

where the first order part vanishes because of the random nuclear dis-

placements and the second order term remains. However, the second

order Doppler shift is quite small, as an example, 0.07 mm/s for 57Fe

for a decrease of 100 K [11].

Figure 1.4: Recoil-free emission or absorption of γ-rays when the nuclei arein solid matrix.

Rudolf L. Mossbauer observed that the recoilless emission and ab-

sorption of γ rays is possible if nuclei are embedded into a solid envi-

ronment (like shown in Figure 1.4) and the transition lines can over-

lap, thus resulting in the resonance absorption. This can be explained

from the classical point of view, when the γ-ray is emitted by a nu-

cleus bound in a lattice, the entire crystal lattice will absorb the recoil.


In this case, the mass M in the denominator of Eqn. 1.1 should be the

mass of the whole crystal, not the individual nucleus. This reduces

the recoil energy to a negligible amount. Therefore, Eγ = E0 and the

entire process becomes a recoilless resonant absorption [5, 9].

Figure 1.5: Energy levels separated by ~ω of a Debye solid.

The quantum mechanical description of the recoil effect is some-

what more complicated. Of course, the lattice vibrations are quan-

tised and the energy can be absorbed or emitted by the crystal lattice

in quanta of certain energy (phonons). For a Debye solid with one vi-

brational frequency ω, the lattice can only receive or release energies

in integral multiples of ~ω. The energy distribution defined by the

population of the levels spaced by ~ω is shown schematically in Fig-

ure 1.5. If ER <~ω, the lattice cannot absorb the recoil energy, i.e., the

zero phonon process occurs, and the γ-ray is emitted without a recoil.

It suggests that there must be zero-phonon transitions ie, emission

process without excitation of phonons in the lattice. There is a cer-

tain probability f (known as Debye-Waller factor or Lamb-Mossbauer

factor) that no lattice excitation (zero-phonon processes) takes place

during γ-emission or absorption whereas f denotes the fraction of nu-

clear transitions which occur without recoil and only for this fraction

1.1. Introduction 9

is the Mossbauer effect observable. This recoil-free fraction can be ex-

pressed as, f = exp (-ER/~ω). Within the Debye model for solids, f

increases with the decreasing transition energy Eγ , with decreasing

temperature, and with increasing Debye temperature θ. Thus, θ is a

measure of the strength of the bond between the Mossbauer atom and

the lattice.

Figure 1.6: The partial overlap of emission and absorption lines due to thedifferent electron-nuclear interactions in the source and absorber.

The recoilless resonant absorption is necessary for the maximum

overlap of the emission line and absorption line. In-order to make the

nuclear resonance absorption of γ-rays successful, the emission and

absorption lines should coincide or at least partially overlap. How-

ever, the complete overlap between the emission and absorption lines

is possible only if identical materials are used as source and absorber.

If the source and the absorber nuclei are in different chemical environ-

ment, which is usually the case, they have slightly different absorp-

tion/emission frequencies due to the interactions with the surround-

ing electrons [5, 9]. This is schematically shown in Figure 1.6, where

the partial overlap of the absorption and emission lines is because of

the different electron nuclear interactions in the absorber and emit-

ter. Because the Mossbauer lines are very sharp, even small energy


differences will destroy the resonance. However, with the use of the

Doppler effect, ie, by moving the source and absorber relative to each

other, the perfect overlap can be obtained. For 57Fe, Doppler veloci-

ties up to a few mm/s are sufficient to achieve good overlap between

the emission and absorption lines. A Mossbauer spectrum, which is a

plot of the relative transmission of the gamma radiation as a function

of the Doppler velocity, reflects the nature and strength of the hyper-

fine interactions between the Mossbauer nucleus and the surrounding

electrons. The Mossbauer effect makes it possible to resolve the hyper-

fine interactions and provide information on the electronic structure.

The three main hyperfine interactions corresponding to the nuclear

moments are,

1) Isomer Shift

2) Quadrupole Splitting

3) Magnetic Hyperfine Splitting

1.1.1 Isomer Shift

Mossbauer Isomer Shift (MIS) is defined as a displacement of the fre-

quency of the nuclear γ-transition in the target (absorber) nucleus ∆Eaγ

with respect to the reference (source) nucleus ∆Esγ [5,6,12]. The varia-

tion of the nuclear volume, ie, the nuclear charge radius, during the γ

transition is responsible for the occurrence of Mossbauer isomer shift,

because the atomic nucleus is not a point-like object but an object of

a finite spatial extent. The different nuclear charge radius in the ex-

cited and ground state induce different electron-nuclear interactions

therein, hence the frequency of the γ transition in the nucleus im-

mersed in a specific electronic environment is different than in the

bare nucleus. In a Mossbauer experiment one measures the change

1.1. Introduction 11

of the energy of the resonance γ quantum between the source (s) and

the absorber (a) nuclei, thus there appears a dependence of the en-

ergy of resonance γ quantum on the electronic environment in which

the given nucleus is immersed. The MIS, δ measured in terms of the

Doppler velocity necessary to achieve resonance is given in Eqn. 1.4,

δ =c

Eγ

(∆Eaγ −∆Es

γ) (1.4)

where c is the velocity of light and Eγ is the energy of the γ quantum.

Figure 1.7: The isomer Shift and Quadrupole Splitting of the nuclear energylevels and corresponding Mossbauer spectra.

This shift appears in the spectrum as the difference between the

position of the baricenter of the resonance signal and zero Doppler

velocity as shown in Figure 1.7 [5, 6, 12]. Traditionally, the energy dif-

ferences ∆Ea/sγ are calculated within the framework of perturbation


theory, whereby the variation of the electron-nuclear interaction po-

tential during the γ-transition is treated as a weak perturbation of the

nuclear energy levels [5, 6, 12–14]. This approach leads to the well

known expression for the isomer shift of Mossbauer spectra as a lin-

ear function of the so-called contact electron density (electron density

at the nucleus) in the absorber ρae and source ρse compounds, see Eqn.

1.5,

δ = α(ρ(a) − ρ(s)) (1.5)

where α is a calibration constant, which depends on the parameters

of the nuclear γ-transition. The most valuable information derived

from isomer shift data refers to the oxidation state and spin state of a

Mossbauer-active atom, its bond properties etc.

1.1.2 Quadrupole splitting

Quadrupole splitting in the Mossbauer spectrum occurs when a nu-

cleus with an electric quadrupole moment experiences a non-uniform

electric field [5, 9]. The nuclear charge distribution deviates from the

spherical symmetry for a nucleus that has spin quantum number I >

1/2 and thus has a non zero electric quadrupole moment. The mag-

nitude of the quadrupole moment may change in going from one

state of excitation to another. The sign of the electric quadrupole

moment, Q indicates the shape of the deformation. Q is negative

for a flattened (pancake-shaped) nucleus and positive for an elon-

gated nucleus (cigar-shaped). Q is constant for a given Mossbauer

nucleus, ie, changes in the quadrupole interaction energy observed

in different compounds of a given Mossbauer nuclide under constant

experimental conditions can only arise from the changes in the elec-

tric field gradient (EFG) generated by the surrounding electrons and


other nuclei. Therefore, the interpretation of quadrupole splittings

requires the knowledge of the EFG. The interaction between the elec-

tric quadrupole moment of the nucleus and EFG at the nuclear po-

sition give rise to a splitting in the nuclear energy levels into sub

states, which are characterised by the absolute magnitude of the nu-

clear magnetic spin quantum number |mI |.

As the Mossbauer spectroscopy involves the absorption of the γ-

rays to promote a nucleus from the ground state to an excited state,

the quadrupole Hamiltonian has to be solved for each energy level if

both levels have nuclear spin greater than 1/2. For 57Fe, the ground

state has nuclear spin I = 1/2 and the lowest excited state has I = 3/2.

The second part of Figure 1.7 shows the quadrupole splitting of the

nuclear energy levels of 57Fe, where the absorption line is split due to

the interaction of the nuclear quadrupole moment with non-zero EFG

at the nucleus. The separation between the lines , ∆EQ, is known as

the quadrupole splitting and is written as,

∆EQ =1

2qQVzz

(

1 + η2

3

)1/2

(1.6)

where e is the electrical charge, Q is the nuclear quadrupole moment,

and V is the electric field gradient due to the total electron density

plus all nuclear charges. V can be decomposed into three principal

components, Vzz, Vyy, and Vxx, in descending order of magnitude, and

η is the asymmetry parameter defined as (Vxx - Vyy)/Vzz. For the

substates with axially symmetric EFG (η = 0), the energy separation

∆EQ is,

∆EQ =1

2qQVzz (1.7)

The quadrupole splitting provides information on the symmetry of

the coordination sphere of the resonating atom.


1.1.3 Magnetic Hyperfine Splitting

The dipole interaction between the nuclear spin moment and the mag-

netic field is called the Magnetic Hyperfine Splitting (Nuclear Zeeman

effect) [5, 9]. A nuclear state with spin I > 1/2 possesses a magnetic

dipole moment µ. The magnetic field splits the nuclear level of spin

I into (2I + 1) equispaced non-degenerate substates characterised by

the magnetic spin quantum numbers mI . Therefore for 57Fe, the ex-

cited state with I = 3/2 is split into four, and the ground state with I

= 1/2 into two substates as shown in Figure 1.8. The energies of the

sublevels are given from first-order perturbation theory by,

EM (mI) = −µHmI/I = −gNβNHmI (1.8)

where βN is the nuclear Bohr magneton, µ is the nuclear magnetic mo-

ment, mI is the magnetic spin quantum number and gN is the nuclear

g-factor.

The magnetic hyperfine splitting enables one to determine the ef-

fective magnetic field acting at the nucleus. The total effective mag-

netic field is the vector sum of externally applied magnetic filed and

the internal magnetic field, ~Heff = ~Hext + ~Hint. The latter consist of

three parts, ~Hint = ~HL + ~HD + ~HC . ~HL is the contribution from the

orbital motion of the electrons, ~HD is the contribution of the magnetic

moment of the spin of the electrons outside the nucleus (spin-dipolar

term) and ~HC is the contribution of the spin-density at the nucleus

(Fermi contact term).

The magnetic hyperfine interaction gives a clear understanding of

the magnetic properties of materials. In compounds with unpaired

electrons the Mossbauer spectroscopy enables one to distinguish be-

tween the high-spin and low-spin states, spin density at various nuclei

1.2. Applications of Mossbauer Spectroscopy 15

Figure 1.8: The Magnetic Splitting of the nuclear energy levels and corre-sponding Mossbauer spectrum.

in a molecule, study the magnetic ordering, etc.

1.2 Applications of Mossbauer Spectroscopy

The fields of applications of Mossbauer spectroscopy includes solid-

state physics / chemistry, bio-chemistry / physics, catalysis, nano-

science, materials science, metallurgy etc. This technique has even

become established for the planetary exploration on the surface of

Mars, the presence of water on Mars is confirmed by Mossbauer spec-

troscopy [15]. Even though there are numerous studies on Mossbauer

active nuclei like Sn, Au, Hg, I, etc., the best studied Mossbauer active

nucleus is 57Fe and we will survey a few applications of Mossbauer

spectroscopy to this element. For a complete review of applications of


Mossbauer spectroscopy, see Refs. [16–18].

Soon after its discovery, Mossbauer spectroscopy was used to solve

problems in solid state research. Fluck et. al [19] successfully used

this technique is to distinguish between Prussian Blue (PB) and Turn-

bull’s Blue (TB). Both these are compounds with the general molecular

formula AxMa[Mb(CN)6]zH2O (where, A = alkali cation and Ma/Mb

= metal ion). PB is prepared by adding FeIII salt to a solution of

[FeII(CN)6]4−, and TB by adding FeII salt to a solution of [FeIII(CN)6]

3−.

It was believed for a long time that these were chemically different

compounds, Prussian Blue with [FeII(CN)6]4− anions and Turnbulls

Blue with [FeIII(CN)6]3− anions. However, the Mossbauer spectra reco-

rded by Fluck et. al [19] were nearly identical for both PB and TB

showing only the presence of [FeII(CN)6]4− and FeIII in the high spin

state. The explanation is that during the preparation of TB, imme-

diately after adding a solution of FeII to a solution of [FeIII(CN)6]3−, a

rapid electron transfer takes place from FeII to the anion [FeIII(CN)6]3−

with subsequent precipitation of the same material as of PB. It is now

agreed that TB and PB are the same because of the rapidity of electron

exchange through a Fe-CN-Fe linkage.

Thermal spin transition (spin crossover) and electron transfer re-

sulting in valence tautomerism are another aspects which attracted

increasing attention by chemists and physicists because of the promis-

ing potential for practical applications in sensors and display devices

[20]. Prussian Blue Analogues are demanding among those compounds

which show pressure and temperature induced electron transfer [21].

For instance, the CoII- FeIII cyano complex when doped with potas-

sium ions, K0.1Co4[Fe(CN)6]2.718 H2O, may undergo a thermally in-

duced electron transfer around 20 K from the CoII with spin S=3/2 to

the FeIII with spin S=1/2, turning the CoII-HS to CoIII-LS and the FeIII-

1.2. Applications of Mossbauer Spectroscopy 17

LS to FeII- LS [21]. The diamagnetic pair CoIII-FeII (total spin S = 0) can

be converted back to the paramagnetic pair with the application of

light. It has been observed that the formation of the diamagnetic pairs

can be enhanced by increasing the potassium concentration or doping

with the alkali ions like rubidium or caesium and with the application

of pressure. Ksenofontov et. al. [21] proved this by measuring the

Mossbauer spectra under applied pressure on a sample which con-

tained a small fraction of potassium. There is no thermally induced

electron transfer at ambient pressure, at 4.2 K, and the spectrum is

magnetically split into a sextet with a local magnetic field of 165 kOe

arising from the S=1/2 Fermi contact field. At a pressure of 3 kbar,

most of the sextet intensity disappears and a singlet arising from the

FeII-LS sites emerges. At 4 kbar, the pressure-induced electron trans-

fer from CoII-HS to FeIII-LS is complete and the spectrum shows only

the typical FeII-LS singlet [21].

Recent discovery of high-temperature superconductivity (HTSC)

in iron-based compounds has initiated a considerable research activ-

ity comparable or even in excess to the discovery of HTCS in cuprates

[22]. Iron-based superconductors belong to pnictide (compounds of

group V elements) [23–25] or chalcogenide (compounds of group VI

elements) type of compounds [26]. In resolving the origin of supercon-

ductivity in these compounds, the knowledge of their local geometry

and local electronic structure is extremely important. Although the

X-ray diffraction methods can provide the reliable crystalline struc-

tures and the long range magnetic order can be studied by the neu-

tron diffraction method, the local geometric and electronic structure is

accessible via the use of Mossbauer spectroscopy, which is capable of

providing information on an atomic scale [5, 27].

Mossbauer spectroscopy technique finds numerous applications in


biological systems. For example, Schunemann et. al [28] have stud-

ied the spin distribution and iron oxidation states of the intermediate

states in the reaction cycle of cytochrome P450. The enzyme super-

family cytochrome P450 are found in many living organisms, and play

an important role in many physiological processes for example in the

biotransformation of xenobiotics and synthesis of steroid hormones.

The active site of cytochrome P450 contains a substrate binding site

next to the heme iron centre. They catalyse a variety of reactions by

the transfer of an active oxygen from its heme unit to the substrates.

When a substrate binds to the active site of the enzyme, intermediate

states are formed, however little is known about these intermediate

states. Mossbauer spectroscopy can give clear evidence about the iron

oxidation states of these intermediate states [28].

The advent of third-generation synchrotron radiation sources ex-

tends the applicability of the Mossbauer technique. The use of syn-

chrotron source is an alternative to the conventional Mossbauer tech-

nique [1, 2]. The radiation from the synchrotron source is intense,

tuneable in energy, and is available in the form of short pulses. There

are several ways of Mossbauer filtration of synchrotron radiation (MFSR),

such as pure nuclear reflection, total external reflection (TER), forward

scattering (nuclear resonant forward scattering (NFS) and nuclear in-

elastic scattering (NIS)) etc. The use of synchrotron radiation over-

comes some of the limitations of the conventional technique. For in-

stance, NFS allows the direct determination of the Lamb-Mossbauer

factor [10]. In addition, the high brilliance and the extremely colli-

mated beam lead to a large flux of photons through the very small

size of the sample (0.1-1 mm2), makes possible to measure extremely

small samples, and also samples under unusual conditions like high

pressure [1, 2].

1.3. Significance of the present work 19

1.3 Significance of the present work

The parameters of Mossbauer spectra, such as the isomer shift, quadru-

pole splitting, magnetic hyperfine splitting, are sensitive characteris-

tics of the electronic structure and carry important information on the

spin- and oxidation-state of the resonating atom as well as on its lo-

cal chemical environment. The theoretical estimation of Mossbauer

isomer shifts and the evaluation of the nuclear structure parameters

therein, is the main goal of this thesis.

The Mossbauer isomer shift is the measure of the energy differ-

ence between the energies of γ-transitions occurring in the absorber

and source. Because the electronic environments in which the sample

and the reference nuclei are immersed are different, the isomer shift

probes this difference. However, the relationship between the isomer

shift and the local electronic structure is not straightforward. The iso-

mer shift depends on the nuclear structure parameters, such as the

charge radius variation during the γ-transition, as well as on the lo-

cal electronic structure parameters, such as the electron density ρa/se

in the vicinity of nucleus [6, 12]. Although measurable in principle,

these characteristics are not directly accessible from the experiment.

In such a situation, the first principles quantum chemical calculations

of the local electronic structure and of the isomer shift are of the ut-

most importance. Thus, the theoretical calculation of the electron con-

tact density at the target nucleus in chemical compounds, and calibra-

tion against the experimental isomer shifts remains the most reliable

way of determining the nuclear structure parameters. In these calcu-

lations, all relevant effects, such as the effects of relativity, the effects

of electron correlation and of the solid-state environment, have to be

included, which makes such calculations very demanding.


The success of the traditional approach for MIS calculation relies

on the availability of the contact densities from the theoretical cal-

culations. The contact density is easily available from the calcula-

tions in which the theoretical methods fulfilling Hellmann-Feynman

theorem, such as the self-consistent field (SCF) method or the Kohn-

Sham (KS) method of density functional theory (DFT), are employed

[29–31]. However, the use of the most sophisticated methods of the

ab initio wave function theory, the coupled-cluster method or single-

reference and multi-reference Møller-Plesset perturbation theory, re-

quires the calculation of the so-called relaxed density matrix which

considerably increases the amount of computational work necessary

to obtain the density. Hence relatively low-level computational meth-

ods are commonly employed in the calibration of the Mossbauer iso-

mer shift.

The standard model of MIS calculations is based on certain as-

sumptions, such as constant density inside the nucleus and point char-

ge nuclear model [6, 12]. The former is valid only within the non-

relativistic formalism, where ρ is replaced with the electron density at

the nuclear position ρ(0). However, the electronic wavefunction in the

vicinity of the nucleus is considerably modified by relativity [12, 32],

hence the relativistic density is divergent near the nucleus and cannot

be defined as ρ(0). Within the standard approach, the non relativistic

contact densities ρ(0) calculated are scaled using an element specific

constant S(Z), in order to account for the effect of relativity. Gen-

erally, this scaling factor is obtained through the comparison of the

four-component relativistic electron density for a one-electron atom

with the corresponding non relativistic density [6]. From the values

of the scaling factor for various elements it is clear that the effect of

relativity is significant in the contact density [6]. Another limitation

1.4. Scope of the thesis 21

is the assumption of point-charge nuclear model. The effect of the

finite size of the nucleus is considered as a perturbation and the per-

turbation Hamiltonian is obtained as the difference between the po-

tential of a uniformly charged sphere and the usual Coulomb poten-

tial [6, 12–14, 18].

Taking all these limitations into consideration, it is desirable there-

fore to employ a more direct theoretical approach for the calculation

of MIS, which allows the straightforward inclusion of the most impor-

tant effects such as relativity and electron correlation. This thesis ex-

plains the development and calculations based on such an approach,

which is based on the non-perturbative inclusion of the finite size nu-

cleus into theoretical calculations. Within this new approach, the most

accurate ab initio methods can be employed, which offers a possibility

for a systematic improvement of the results of theoretical modelling

of the Mossbauer isomer shift.

1.4 Scope of the thesis

In chapter 2, all the quantum chemical formalisms within the scope of

this thesis are introduced. The inclusion of relativity via Normalized

Elimination of Small the Component method in the calculations are

explained. In this chapter, the traditional approach for the MIS calcu-

lations and its limitations are explained, describing the necessity of a

more accurate theoretical framework. The new approach, according

to which the isomer shift is treated as the linear response of the elec-

tronic density with respect to the nuclear radius is described in detail.

Chapter 3 validates density functional theory for the calculation

of MIS within the context of the new theoretical formalism. The val-


idation of pure and hybrid density functionals for the modelling of

Mossbauer Isomer shift is illustrated. The investigation of the effect

of basis set truncation on the calculated values of isomer shifts are

carried out and described in detail.

In Chapter 4, the importance of the proper account of relativity and

electron correlation is demonstrated in a series of atomic calculations

for various oxidation states of iron atom. From the Mossbauer exper-

iments the parameters of the nuclear γ-transition are not known with

sufficiently high accuracy, hence the calibration of the theoretically ob-

tained contact densities with the experimental isomer shifts is impor-

tant (see Eqn. 1.5). The calibration constant α in Eqn. 1.5 gives infor-

mation on the internal parameters of the nuclear γ-transition, such as

the variation of the nuclear charge radius. Chapter 4 describes the cal-

ibration of 119Sn contact densities with the experimental isomer shifts

to obtain a reliable value of the calibration constant α. The obtained

value of calibration constant α is validated by the comparison of cal-

culated vs. experimental isomer shifts. An independent test of the

calibration constant is carried out by studying the isomer shift varia-

tion under pressure for CaSnO3 perovskite.

Chapter 5 details the calculation of 57Fe calibration constant α us-

ing a series of iron compounds. The reliability of the obtained cali-

bration constant, α is tested via calculating the relative isomer shifts

∆δx = δx − δref , choosing various reference compounds.

Chapter 6 describes the investigation on the possible origin of two

different signals in the Mossbauer spectra of Prussian blue analogue

RbMnFe(CN)6H2O, in which a switching of magnetism occurs with

light and temperature. The analysis is based on two parameters, i)

the distribution of Rb+ around iron and ii) the difference in oxidation

states of iron.

1.4. Scope of the thesis 23

Chapter 7 details the study on the recently discovered iron based

superconductors [22–26]. The aim is to analyse the influence of the

structural variation near the 57Fe nucleus and chemical substitutions

in the coordination sphere of 57Fe, on the isomer shift and magnetic

hyperfine coupling constants in these compounds. This will give us an

idea on the applicability of Mossbauer parameters to understand the

structure of these iron based superconductors under phase transitions

and upon chemical substitutions.

Chapter 8 give a conclusion and outlook of the dissertation.

Chapter 2

Theoretical Framework

Synopsis

This chapter describes the methods used to calculate the properties within

this dissertation. The details of the quantum chemical methods used in the

calculations and the inclusion of relativity via the Normalized Elimination of

the Small Component method are described. The new theoretical formalism,

according to which the Mossbauer Isomer Shift is treated as a derivative of

the electronic energy with respect to the nuclear charge radius is explained.

2.1 General quantum mechanical formalism

In this chapter all the methods used for the research carried out within

this thesis are explained briefly. For a more in-depth review of the

quantum chemical methods, a number of references are available [33–

36].

2.1.1 Schrodinger equation

The goal of computational chemistry is to solve an eigenvalue equa-

tion of the following form,

HΨ = EΨ (2.1)

26 2. Theoretical Framework

where Ψ is an N-body wavefunction and E is the energy of either the

ground or an excited state of the system. A common and very rea-

sonable approximation used in the solution of Eqn. 2.1 is the Born-

Oppenheimer Approximation. In a system of interacting electrons

and nuclei, the nuclei (being much heavier than the electrons) move

relatively slowly which suggests that it would be reasonable to sepa-

rate the two types of motion. This is the basis of the Born-Oppenheimer

approximation (or separation), according to which the Hamiltonian

(Eqn. 2.1) can be written as,

H =∑

i

−1

2∇2

i +∑

i

∑

α

Zα

riα+

1

2

∑

i

∑

j 6=i

1

rij

+1

2

∑

α

∑

β 6=α

ZαZβ

rαβ

(2.2)

where i and j refer to the electrons while α and β refer to the nu-

clei, r denotes the spatial positions and Z is the nuclear charge. The

first term in Eqn. 2.2 is the kinetic energy term, the second term is the

electron-nuclear attraction term, the third term is the electron-electron

repulsion, the fourth term is nuclear-nuclear repulsion. All proper-

ties can be calculated from the wavefunction, once the equations are

solved. For most physical problems of interest, a number of inter-

acting electrons and ions are involved. However the total number

of particles is usually sufficiently large and the exact solution cannot

be found. In-order to get a qualitatively correct solutions to the many

body Scrodinger equation simplifying approximations are needed. Th-

ese approximations are discussed later in this chapter.

2.1. General quantum mechanical formalism 27

2.1.2 Dirac Equation and importance of relativistic ef-

fects

The theory of special relativity was developed by Einstein in the be-

ginning of the last century and extended by Dirac to the quantum do-

main. It adds a correction to classical physics which becomes impor-

tant when particles move at velocities close to the speed of light (rela-

tivistic velocities). The relativistic effects on energies and other phys-

ical quantities increases with the fourth power of the nuclear charge

Z, hence it becomes crucial for the calculations involving heavy ele-

ments. The importance of relativity for the electronic structure cal-

culations is briefly explained here. For a complete review see Refs.

[37, 38].

Direct relativistic effects

The relativistic effects in atoms and molecules can be divided into two,

(i) the effects which do not cause a splitting of energy levels due to the

spin degrees of freedom and, (ii) the effects of spin-orbit coupling.

The former effects are experienced by the electrons moving with high

velocity in the vicinity of the nucleus, leading to a contraction of the

orbitals and electron-density distribution. This can be explained using

Bohr model of orbits. According to Bohr model, the radius of an orbit

is,

a0 =~

mecα(2.3)

where me is the electron rest mass, α is the fine structure constant, c is

the speed of light (c = 137.0359895 a.u.). The mass of a moving electron


me(ve) is related to the rest mass me as,

me(ve) =me

√

1− v2e/c2)

(2.4)

where the speed of the electron ve ∝ Z/n, Z is the nuclear charge and

n the principal quantum number. Therefore, heavy elements have the

fastest electrons which are in the core orbitals (s and p orbitals). Thus,

Eqn. 2.3 becomes,

arel0 =~√

1− v2e/c2

mecα⇒ a0

√

1− v2e/c2 (2.5)

This explains the contraction of the s and p1/2 orbitals and connected

with this contraction there is a lowering of orbital energies. This direct

influence is termed direct relativistic effects.

Indirect relativistic effects

In addition to the primary effects of relativity on the electronic struc-

ture, there are secondary effects arising from the operation of primary

effects in a many electron environment. As discussed earlier, relativ-

ity is most important for the core orbitals which have high effective

charge and small principal quantum number. The contracted s and

p1/2 orbitals screen the nucleus more effectively resulting in an expan-

sion of the d and f orbitals. These effects are called indirect relativistic

effects.

Dirac Equation

The inclusion of relativistic effects into the quantum chemical calcula-

tions can be achieved through the use of the Dirac equation, which is


valid for the elementary spin-1/2 particles,

−i∂ψ(r, t)

δt= HDψ(r, t) (2.6)

where the Dirac Hamiltonian is,

HD = cα.p+ βmc2 (2.7)

c is the speed of light, m the rest mass of the electron, the standard

definition of the momentum operator is, p = -i ∇, α and β are the 4×4

matrices which are in the usual form,

β =

(

I2 0

0 −I2

)

, αx =

(

0 σx

σx 0

)

, αy =

(

0 σy

σy 0

)

, αz =

(

0 σz

σz 0

)

.

the σ’s are the 2×2 Pauli matrices, I2 and 0 are the 2×2 identity and

null matrix respectively.

In the Dirac equation, the eigenvalues are usually shifted by the elec-

tron rest mass, that is, ε = (E - mc2). Thus the Dirac equation, a set of

coupled differential equations which can be written as,

ε 0 −cpz −c(px − ipy)

0 ε −c(px + ipy) cpz

−cpz −c(px − ipy) ε+ 2mc2 0

−c(px + ipy) cpz 0 ε+ 2mc2

ψ1

ψ2

ψ3

ψ4

= 0(2.8)

Thus the Dirac Hamiltonian can be expressed as a 4×4 matrix which

operates on the wavefunction ψ. This means that the solution leads

to four-component wavefunctions or spinors. The four-component

wave function can be split into two two-component wave-functions:

large component ψL and small component ψS . The Dirac equation

admits both positive energy solutions associated with electrons (ε ∈


[-mc2, +∞]) and negative energy solutions associated with positrons

(ε ∈ [-∞, -2mc2]). These electronic and positronic states are superpo-

sitions of ψL and ψS . ψL determines the electronic states and it ap-

proaches asymptotically the non-relativistic wavefunction for v << c.

Even though, both components, ψL and ψS , are important for the elec-

tronic wavefunction, the four-component relativistic quantum chemi-

cal methods are computationally expensive [39,40]. However in chem-

ical applications, the solutions of interest are the positive energy states.

There are several schemes to reduce the four-component Dirac equa-

tion to a two component equation. This can be done by decoupling the

large and the small components and make the transformed Hamilto-

nian block diagonal, in different ways, the Foldy-Wouthuysen trans-

formation, Douglas-Kroll transformation, Normalized Elimination of

the Small Component (NESC), Regular Approximation etc. Among

these methods, Normalized Elimination of the Small Component (NESC)

[41] method is an efficient computational protocol for obtaining the

exact positive-energy eigenvalues, which is used for the calculations

within this thesis.

Normalized Elimination of the Small Component

Normalized Elimination of Small Component (NESC) [41] is a simple

and efficient computational protocol for obtaining the exact positive-

energy eigenvalues of the relativistic Hamiltonian starting from the

energies obtained within the regular approximation [41]. This method

has the advantages that it can lead to stable and quick convergence

to the exact energies, easy to implement in non-relativistic quantum-

chemical codes, many-electron case can be dealt with one-electron ap-

proximation made in the Hamiltonian.


The expansion of the one-electron wavefunction in the modified

Dirac equation [41] in terms of basis set functions χ according to the

equation, ψi = |χ >Ci, where |χ > is the row vector of basis functions

and Ci is the column vector of expansion coefficients, obtains the ma-

trix modified Dirac equations given by [43],

TB+VA = SAE (2.9)

TA+ (W0 −T)B =1

2mc2TBE

where A and B are the matrices of the expansion coefficients for the

large and small components, T and V are the matrices of the kinetic

and potential energy operators, S is the overlap matrix, and W0 is the

matrix of the operator (σ.p)V(σ.p)/(4m2c2) [43].

The elimination of the small components in Eqn. 2.9 is obtained with

the use of a general nonunitary transformation matrix U, which con-

nects the expansion coefficient matrices A and B as [43],

B = UA (2.10)

The elimination of the small component is done, retaining the proper

nomalization of the wavefunction,

A†SA+

1

2mc2B

†TB = I, (2.11)

one obtains the NESC equation,

(T+ V)A = SAE (2.12)

where the modified kinetic energy, potential energy, and overlap ma-

trices are defined as,

T = U†T+TU−U

†TU (2.13)

V = V +U†W0U

S = S+1

2mc2U

†TU


The matrix U satisfies the equation,

U = T−1(SS−1

H−V) (2.14)

The Eqns. 2.12 and 2.14 are solved iteratively starting with a suit-

able guess for U. Thus the NESC method corresponds to the projection

of Dirac Hamiltonian into a set of positive-energy eigenstates, which

guarantees the variational stability. This approach is extended to the

many-electron case using the one-electron approximation in the rela-

tivistic transformation operators. According to this approximation, it

is only the nuclear attraction potential that is used in the relativistic

transformations [42]. The two-electron terms would make only small

contribution to the transformed Hamiltonian and can be neglected

without considerable loss of accuracy, which reduces the computa-

tional complexity.

2.1.3 Hartree Fock theory

Hartree Fock theory (HF) is one of the simplest approximate theo-

ries for solving the many-body Hamiltonian, Eqn. 2.2. The solu-

tion is complicated if the electron-electron repulsion term, as given

by the third term in Eqn. 2.2, is included in the electronic structure

calculations. The many-electron Hamiltonian is inseparable because

of the electron-electron repulsion term. In atoms and molecules, the

electron-nuclear attraction is stronger than the electron-electron repul-

sion. As a crude approximation, one can neglect the electron-electron

repulsion. Thus, the Hamiltonian becomes separable and the wave-

function becomes a product of one-electron orbitals. This is the inde-

pendent particles approximation, in which it is assumed that the elec-

trons move independently from one another. The N-electron wave-


function for a given state is thus written as a Slater determinant, that

is an anti-symmetrized product of spin orbitals; products of spatial

and spin functions,

Ψ0(x;R) = (n!)−1/2det|φa(1)φb(2)....φz(n)| (2.15)

where the spin orbitals φu, with u = a,b,...,z, are orthonormal and the

label u incorporates the spin state as well as the spatial state.

In the HF method, the determinantal wavefunction is used to ap-

proximate the exact wavefunction and the energy is calculated as an

expectation value of the Hamiltonian over this approximate wave-

function. The orbitals are found using the variational principle by

minimization of the energy expectation value. Thus, the best approx-

imate wavefunction can be obtained by varying all the wavefunction

parameters, until the energy expectation value of the approximate

wave function is minimised. The application of this minimising pro-

cedure leads to the Hartree-Fock equations for the individual spinor-

bitals. The Hartree-Fock equation for spinorbitals φa(1), where elec-

tron 1 is arbitrarily assigned to spinorbital ψa is,

f1φa(1) = ǫaφa(1) (2.16)

where ǫa is the spinorbital energy and f1 is the Fock operator:

f1 = h1 +∑

u

(Ju(1)−Ku(1)) (2.17)

where h1 is the core Hamiltonian for electron 1, the sum is over all

spinorbitals u = a,b,...,z, Ju and Ku are the classical Coulomb and the

exchange operators respectively. The Coulomb operator takes into ac-

count the Coulombic repulsion between electrons, and the exchange

operator represents the quantum correlation due to the Pauli exclu-

sion principle. The sum in Eqn. 2.17 represents the average potential


energy of electron 1 due to the presence of the other (n− 1) electrons,

ie, the sum includes contributions from all spinorbitals φu, except the

φa which is being computed.

The HF equations are solved by using self-consistent field (SCF)

approach, in which a trial set of spinorbitals are guessed and used to

construct the Fock operator. The new set of spinorbitals obtained from

the solution are used to obtain a revised Fock operator, and this cycle

of calculation and reformulation is repeated until the convergence cri-

terion is satisfied. The Hartree-Fock calculations scales as the fourth

power of the number of basis functions used.

2.1.4 Electron correlation treatment

In the HF method, the electrons are assumed to move independently

from each other. The energy contribution resulting from the corre-

lated motion of electrons is usually denoted as the correlation energy

and is defined as the difference between the exact energy and the en-

ergy obtained by the Hartree-Fock method ie, Ecorr = Eexact - EHF . To

take into account the electron-electron interaction, two different ap-

proaches can be used: configuration interaction (CI) and perturbation

theory (Coupled Cluster methods and Møller Plesset Perturbation the-

ory methods).

Generally, the electron correlation is treated as a small correction

∆Ψ to the independent particles model Ψ0,

Ψexact = Ψ0 +∆Ψ (2.18)

where ∆Ψ can be expanded in terms of the wavefunctions orthogonal

to Ψ. The simplest choice of these wavefunctions is the functions ob-

tained by electron excitations from Ψ0. Using the HF wavefunction as


Ψ0, this leads to,

Ψcorr = C0ΨHF +∑

Cai ψ

ai +

∑

Cabij ψ

abij + ...... (2.19)

where ΨHF is the ground-state HF wavefunction, ψai is the singly ex-

cited Slater determinant, ψabij is the doubly excited Slater determinants,

etc.. Note that the virtual orbitals are chosen to be orthogonal to the

occupied ones. The coefficients in this expansion can be found either

using the variational principle or perturbation theory.

Within the variational principle (configuration interaction (CI) ap-

proach), the wavefunction is expanded in a set of determinants where

the electrons have been separated by excitations. A singly excited

determinant corresponds to one for which a single electron in occu-

pied spinorbital ψi has been promoted to a virtual spinorbital ψa and

a doubly excited determinant is one in which two electrons have been

promoted, one from ψi to ψa and one from ψj to ψb and so on (see

Figure 2.1). Thus, the configurations are generated by doing single,

double, etc., excitations from the ground state configuration. A linear

combination of a small number of them constructed to have a correct

electronic symmetry is called a configuration state function (CSF). At

present, such an expansion can contain as many as a billion determi-

nants. Full CI calculations include all possible excitations for a given

molecule in a given basis. These calculations are extremely expensive

and can be done only for the smallest molecules. Truncated CI (CISD

etc) can be used as a simplification, however these methods are not

size-consistent.

The amplitudes of the excited determinants in the many-electron

wavefunction can be obtained with the use of the perturbation the-

ory. The standard Rayleigh-Schrodinger perturbation theory can be

applied up to a finite order, which leads to the Møller-Plesset theory


Figure 2.1: The notation for excited determinants.

based on the HF approximation, or a summation of certain terms in

the perturbational expansion can be carried out up to the infinite or-

der, which leads to the coupled cluster theory.

The single Slater determinant is not a good approximation in cer-

tain cases, e.g. dissociating bonds, open electronic shells, etc. In these

situations, a better approximation can be obtained by considering a

linear combination of several determinants, which leads to a multi-

configurational description.

Coupled Cluster Method

In the CC theory, the summation of the perturbational expansion is

achieved implicitly by using the so-called cluster operator T , which

relates the exact electronic wavefunction Ψ to the HF wavefunction

Ψ0 through,

Ψ = eTΨ0 (2.20)


where the exponential operator eT is defined by the series expansion

eT = 1 + T +1

2!T 2 +

1

3!T 3 + ... (2.21)

The effect of the exponential operator eT on Ψ0 is to create a linear

combination of Slater determinants that includes ψ0 and all its excited

determinants. The cluster operator is given as, T = T 1 + T 2 + .... + TN .

The effects of the excitation operators are,

T1Ψ0 =∑

i,a

taiψai

T2Ψ0 =∑

i,j,a,b

tabij ψabij (2.22)

where tai are called single-excitation amplitudes, tabij double-excitation

amplitudes, etc.

The effect of eT on Ψ0 yields terms of the form T 1 Ψ0, T 2Ψ0, T 3 Ψ0,

..., it also results in products of excitation operators such as T 1 T 1 Ψ0,

T 1 T 2 Ψ0, and T 1 T 2 T 3 Ψ0, where T 2 Ψ0 represents a connected dou-

ble excitation contribution containing double-excitation amplitudes

tabij whereas T 1 T 1 Ψ0 represents a disconnected double-excitation con-

tribution containing products of tai tbj single-excitation amplitudes. In

the Taylor-series expansion, only a finite number of terms will appear.

In coupled cluster singles and doubles (CCSD) approach, the expres-

sion for the cluster operator T is approximated by T 1 + T 2. For a

specified basis set for determination of the spinorbitals, a full CI cal-

culation and a CC calculation including all excitation operators (T 1,

T 2, ..., TN ) would yield identical electronic energies.

Coupled cluster calculations are size extensive but not variational.

By including many excitation terms in the expansion, CC methods are

computationally very expensive relative to HF calculations. Formally,


CC singles-doubles (CCSD) scales asN6 where N is the number of ba-

sis states included in the expansion, calculations including up to triple

excitations scale as N8, and so on. One way to improve this is to in-

clude higher excitations as a perturbation leading e.g. to the CCSD(T)

method, where the unconnected triple excitations are calculated as a

perturbation, it scales as N7. Thus, the key limitation of the CC meth-

ods are their rapid increase in computational cost with system size.

Møller-Plesset Perturbation Theory

The Møller-Plesset perturbation theory uses the difference between

the exact electron-electron interaction and the SCF two-electron en-

ergy, as a perturbation,

H = H(0) + H(1) =

n∑

i=1

fi +

(

∑

i>j

1

rij−∑

i

(Ji − Ki)

)

(2.23)

where the first term gives the zero-order Hamiltonian (H(0)) which is

the sum of one-electron Fock operators and the second term repre-

sents the perturbation. The Hartree-Fock energy is equal to the sum

of the zero-order energy E(0)0 and the first order energy correction E

(1)0 .

Therefore, the first correction to the ground-state energy is given by

the second-order Rayleigh-Schrodinger perturbation theory as,

E(2) =∑

J 6=0

< φJ |H(1)|φ0 >< φ0|H

(1)|φJ >

E(0)0 −E

(0)J

(2.24)

where φJ is a multiply excited determinant and an eigenfunction of

H(0), with eigenvalue E(0)J . The use of the HF wavefunction and the

choice of the zero-order Hamiltonian as a sum of the Fock operators

lead to a compact expression where only the orbital energy differences


enter the denominator.

E(2)C =

1

4

occ∑

i,j

vir∑

a,b

((ij||ab)(ab||ij)

ǫi + ǫj − ǫa − ǫb(2.25)

It is possible to include third and fourth-order energy corrections to

form MP3, MP4 and so on. The MPn energies are size consistent, but

not variational. MP2 scales as the fifth power of the number of basis

states included in the expansion.

Density functional theory

In 1964 Hohenberg and Kohn proved a theorem, that led to one of the

most widely used computational methods [44]. They showed, that

the ground state energy and all ground state properties are uniquely

determined by the ground state electron density. Thus the ground

state energy becomes a functional of the density E[ρ]. This is quite

an intriguing statement. The wave function for an n-electron sys-

tem depends on 3n spatial coordinates, while according to Hohenberg

and Kohn, a function of three spatial coordinates (density) is suffi-

cient. To make this into a useful computational theory, we need the

second Hohenberg and Kohn theorem, the variational theorem. It

states, that the true ground state electron density minimises the en-

ergy functional (similar to the variational method in wave function

theory, which states, that the true wavefunction minimises the en-

ergy). While DFT in principle is an exact method, it delivers only

approximative results, because the kinetic-energy functional and the

electron-interaction functional are not exactly known.

Kohn and Sham [45] developed DFT further and showed that it is

possible to find the exact ground state density from a fictitious system


of non-interacting particles moving in a modified external potential.

The exact energy functional can be written as,

E[ρ] = Ts[ρ] +1

2

∫ ∫

ρ(r)ρ(r′)

|r − r′|drdr′ + EXC [ρ] +

∫

ρVext(r)dr (2.26)

where Ts[n(r)] is defined as the kinetic energy of non-interacting elec-

trons with density ρ,

Ts[ρ] = −1

2

∑

∫

ψ∗i (r)∇

2ψi(r)dr (2.27)

EXC[ρ] in Eqn. 2.26 is the exchange-correlation energy functional and

the ground state density is,

ρ =∑

i

|ψi|2 (2.28)

The Kohn-Sham orbitals Ψi are found as the orbitals which mini-

mize the energy of a system of non-interacting particles moving in a

modified potential,

Veff(r) = Vext(r) +

∫

ρ′

|r − r′|dr′ + VXC(r) (2.29)

where

VXC(r) =δEXC [ρ]

δρ(2.30)

Thus, they are solutions of the one-electron Kohn-Sham equations,

(−1

2∇2

i + Veff(r))ψi(r) = εiψi (2.31)

A self-consistent solution is required due to the dependence of

Veff on n(r). The above equations provide a theoretically exact method

for finding the ground state energy of an interacting system provided


the form of EXC is known. However, the exact EXC is not known, there-

fore one resorts to using various approximations. In electronic struc-

ture calculations EXC is most commonly approximated using the Local

Density Approximation (LDA) or Generalised-Gradient Approxima-

tion (GGA). In LDA, the exchange-correlation energy density εXC[ρ] of

an atom or molecule is replaced by the εXC of a uniform electron gas.

GGA corrects the LDA energy expression for a density inhomogeneity

by introducing a term that depends on the density gradient,

EGGAXC [ρ] =

∫

ǫXC(ρ)ρdr +

∫

FXC [ρ, |∇ρ|]dr (2.32)

where FXC is the correction term. There are numerous exchange-corre-

lation functionals have been developed in the past few decades.

Perdew has formulated the hierarchy of DFT approximations as a

”Jacob’s ladder” [46]. The first rung of this ladder is the local (spin)

density approximation [LDA, e.g., SVWN [47, 48]] and the second

rung is the generalized gradient approximation [GGA, e.g., BLYP [49,

50] and PBE [51]]. The third rung is termed meta-GGA [e.g., TPSS

[52]] and the fourth rung is hybrid DFT that introduces non-locality

by replacing some portion of the local exchange energy density with

the exact (HF-like) exchange energy density [e.g. BH-HLYP [53] and

B3LYP [54]]. The fifth rung of the ladder in which in addition to the

occupied Kohn-Sham orbitals, unoccupied Kohn-Sham orbitals are

utilized leading to methods which converge in a sense to traditional

wavefunction methods and Kohn-Sham versions of coupled cluster

type or second-order perturbation theory (PT2). One of the new hy-

brid density functionals suggested by S. Grimme [55], B2-PLYP, is

based on the mixing of standard generalized gradient approximations

(GGAs), for exchange by Becke (B) and for correlation by Lee, Yang

and Parr (LYP), with Hartree-Fock (HF) exchange and a perturbative


second-order correlation part (PT2) that is obtained from the Kohn-

Sham (GGA) orbitals and eigenvalues.

The expression for the exchange-correlation energy according to

the double hybrid functional is,

Exc = (1− ax)EGGAx + axE

HFx + bEGGA

c + cEPT2c (2.33)

where EPT2c is the usual second-order Møller Plesset type expression

for the correlation energy as in Eqn. 2.25, but here evaluated with the

Kohn-Sham orbitals with the corresponding eigenvalues ǫ. ax is the

HF-exchange mixing parameter and b and c scale the contributions

of GGA and perturbative correlation contributions respectively. The

calculations reported in this thesis are carried out with the use of tra-

ditional density functional approaches based on the LDA and GGA

as well as with the use of the hybrid and double hybrid density func-

tionals.

2.2 Theory of Mossbauer Isomer Shifts

Mossbauer Isomer Shift (MIS) is the shift of the centre of the Mossbauer

spectrum as a result of the Coulomb interaction of the nuclear-charge

distribution with the surrounding electrons. The nuclear charge ra-

dius varies during the nuclear γ-transition which leads to a small vari-

ation of the transition energy that can be sensed using the Mossbauer

effect. The MIS is usually expressed in terms of the Doppler velocity

necessary to achieve the resonance absorption, as given in Eqn. 1.4. In

this section, the evaluation of a new computational formalism to MIS

is introduced.

2.2. Theory of Mossbauer Isomer Shifts 43

2.2.1 Perturbative treatment of Mossbauer Isomer Shifts

The traditional approach to the calculation of Mossbauer Isomer Shift

is based on a number of approximations, such as the point charge nu-

clear model and the constant electron density at the nucleus. However

in reality, the nucleus has a finite size. The finite size of the nucleus

is treated as a weak perturbation within the traditional approach, ie,

the variation of the electron nuclear interaction during the γ-transition

is treated as a weak perturbation of the point charge nuclear model.

Within perturbation theory, the interaction energy is evaluated as the

first order perturbation [5, 6, 12–14], ie,

∆E =< ψ(0)el. |H

(1)|ψ(0)el. > (2.34)

The perturbation Hamiltonian is usually obtained as a difference

between the potential arising from a uniform charge distribution within

the nucleus of finite radius and the potential generated by a point

charge nucleus.

∆H(1) =Z

R

[

3

2−

1

2

( r

R

)2

−R

r

]

, r ≤ R

= 0, r > R (2.35)

where r is the distance from the center of the nucleus, R is the radius

of the uniformly charged sphere. The nuclear charge radius R can be

obtained from the experimental value of the root mean square (RMS)

nuclear radius < R2 >1/2, as,

R =

√

5

3< R2 >1/2 (2.36)

With the assumption of constant density inside the nucleus, one

arrives at the expression for the energy shift as,

∆E =2π

5ZR2ρ (2.37)


Considering the energy correction due to the nuclear finite size to

the energies of the ground and excited states of the nucleus, the energy

shift of the source (absorber) nucleus is,

∆E =2π

5Z((R +∆R)2 − R2)ρ ≈

4π

5ZR2∆R

Rρ (2.38)

Substitution of Eq. 2.38 in Eq. 1.4 gives the MIS connected to the

electron density difference between the source and absorber nuclei.

δ =c

Eγ

4π

5ZS(Z)R2

(

∆R

R

)

(ρa − ρs) (2.39)

where Eγ is the energy of the nuclear γ-transition, c is the velocity of

light, Z andR are the nuclear charge and radius, ∆R is the variation of

the nuclear radius, and ρae and ρse are the average electronic densities

inside the absorber and the source nucleus respectively.

Within this approach to the isomer shift calculation, the electron

density ρ is obtained from the non-relativistic quantum chemical meth-

ods using the point-charge nuclear model. However, the electronic

wave function in the vicinity of nucleus is strongly modified by rel-

ativity [12, 32], hence the electron density ρ is divergent near the nu-

cleus. The non-relativistic densities at the nucleus are corrected with

a relativistic scaling factor S(Z) (see Eqn. 2.39) in order to account for

the relativistic effect on the average electron density inside the nuclear

charge radius. However, the inclusion of relativity using such an ele-

ment specific constant is not sufficient. Another limitation of the tradi-

tional approach is that, the use of high level quantum chemical meth-

ods for the MIS calculations, such as the coupled-cluster method or

single-reference and multi-reference Møller-Plesset perturbation the-

ory, requires the calculation of the so-called relaxed density matrix

(the density matrix which includes the orbital response, Ref. [56]),


which considerably increases the amount of computational work nec-

essary to obtain the density. The use of density matrices without

the orbital response leads to incorrect densities. Furthermore, the re-

laxed density matrix is currently not routinely available for the multi-

reference extensions of the many-body perturbation theory, such as

the complete active space second order perturbation theory (CASPT2)

or the multi-reference coupled cluster methods [57]. Hence relatively

low-level computational methods are commonly employed in the cal-

ibration of the Mossbauer isomer shift using Eqn. 2.39. At this point, it

seems desirable to develop a new theoretical formalism which directly

incorporates the effects of relativity and electron correlation, and al-

lows for the theoretical modelling of the isomer shift within the most

accurate ab initio methods.

2.2.2 Mossbauer Isomer shift as energy derivative

Recently, a new method [58] to the calculation of MIS was suggested.

Within this approach, a connection between the physical origin of the

Mossbauer isomer shift and the origin of the isotope shift of the elec-

tronic energy terms is used [58,59]. The interaction with the nucleus of

a finite size produces the same energy shift for the electronic energy

terms as well as for the nuclear terms. Thus the energy shift of the

γ quantum during the Mossbauer nuclear transition can be written

as [58, 59],

∆Eγ =< Ψe|He(VeNe)|Ψe > − < Ψe|He(V

gNe)|Ψe > (2.40)

where He(VgNe) is the electronic Hamiltonian including the electron-

nuclear interaction potential VgNe . This is the potential of a finite nu-

cleus with the root mean square charge radius of the ground (g) or


the excited (e) state of the nucleus. Assuming that the nuclear charge

distribution is spherically symmetric and the variation of nuclear ra-

dius R during the γ transition is very small (∆R/R ≈ 10−4), one can

express the energy shift as,

∆Eγ =δEe(R)

δR

∣

∣

∣

∣

R=R0

∆R +1

2

δ2Ee(R)2

δR

∣

∣

∣

∣

R=R0

(∆R)2 + .... (2.41)

where Ee(R) is the electronic energy calculated for a nucleus of finite

size specified by the charge radius R, R0 is the experimentally mea-

sured charge radius of the nucleus in the ground state. Thus the en-

ergy shift of the Mossbauer γ transition is defined as the change in

the electronic energy due to the variation of the nuclear radius. It has

been found in Ref. [58] that the dependence of electronic energy on

the nuclear radius is nearly perfectly linear (the effect of non-linearity

is ≈ 0.01 %), see Figure 2.2. Therefore retaining only the lowest order

derivatives in Eqn. 2.41, the Mossbauer isomer shift is expressed as,

δ =c

Eγ

(

δEae (R)

δR

∣

∣

∣

∣

R=R0

−δEs

e(R)

δRN

∣

∣

∣

∣

R=R0

)

∆R (2.42)

where Eae and Es

e are the electronic energies of systems containing the

absorber and the source nuclei.

Even though, the new approach to calculate MIS (Eqn. 2.42) looks

different from the standard perturbational formula (Eqn. 2.39), a straig-

htforward connection between the two approaches can be established

under a number of assumptions such as, (i) the electronic energy is

obtained variationally (that is, that the Hellmann-Feynman theorem

applies), (ii) the nuclear charge distribution is represented by a uni-

formly charged sphere, and (iii) the electron density inside the nu-

cleus is constant. Under these assumptions one obtains Eqn. 2.43 for


Figure 2.2: Total electronic energy of Sn (in hartree a.u.) calculated at theNESC/HF level of theory as a function of the R.M.S. nuclear radius (given in

bohr), taken from Ref. [58].

the derivative of the electronic energy with respect to the nuclear ra-

dius,

δEe(R)

δR

∣

∣

∣

∣

R=R0

=

⟨

Ψe

∣

∣

∣

∣

∣

δHe(VNe)

δR

∣

∣

∣

∣

∣

Ψe

⟩

|R=R0

=

⟨

Ψe

∣

∣

∣

∣

∣

3Z

2R40

∑

i

(R20 − r2i )H0(R0 − ri)

∣

∣

∣

∣

∣

Ψe

⟩

=4π

5ZR0ρ (2.43)

where the index i runs over all the electrons in the system, ri is the dis-

tance between the center of the given nucleus and the ith electron, and

H0(x) is the Heaviside step function, which guarantees that the inte-

gration in the second line of Eqn. 2.43 is carried out inside a sphere of

radius R0 around the nucleus. Substituting Eqn. 2.43 in Eqn. 2.42 one


arrives at Eqn. 2.39. Thus, under the assumptions (i)-(iii), Eqn. 2.42

matches with the standard perturbational approach to the Mossbauer

isomer shift. However, the assumptions (i)-(iii) are not implicit in Eqn.

2.42 and hence the applicability of the method based on Eqn. 2.42 are

not restricted to the methods based on the variational principle.

In practical application of the approach based on Eqn. 2.42, the

derivatives δEa

e(R)

δRare calculated by numeric differentiation of the elec-

tronic energy with respect to the nuclear radius using the increment

of 10−6 Bohr for the root mean square (r.m.s.) nuclear charge radius

(RSn = 0.87694×10−4 Bohr andRFe = 0.70213 × 10−4 Bohr). This is pos-

sible due to the linear dependence of electronic energy on the nuclear

charge radius. Thus, relatively large increments of R can be used. In

this way, the necessity to compute the relaxed density matrix is by-

passed which makes the new approach easy to use in connection with

the non-variational quantum chemical methods, such as the second

order Møller-Plesset perturbation theory (MP2) or the coupled clus-

ter with single and double substitutions and noniterative treatment

of triple excitations CCSD(T) method. Formally, Eqn. 2.42 can be

brought to the form of Eqn. 2.39 if one defines the effective average

electron density inside the nucleus as in,

ρ =5

4πZR

δEe(R)

δR

∣

∣

∣

∣

∣

R=R0

(2.44)

The so-defined effective density ρe can be used as an analog of the

conventional contact density in Eqn. 2.39 (see Table 2.1, the electron

contact densities obtained according to Eqn. 2.44 are given along with

the traditionally obtained densities).

The described approach can be used with any model of nuclear


Table 2.1: Average density inside the nucleus ρ in bohr−3 evaluated fromEqn. 2.44 for the ground states of neutral iron and tin atoms, taken from

Ref. [58].

HF MP2 CCSD(T) Ref.

Fe 14894.186 14895.146 14894.788 15092.04a

(11850.384)b (11850.861) (11848.838) (11903.987)c

Sn 183409.328 183417.235 183414.567 187005.5

(86922.221) (86923.559) (86922.479) .....

aNumeric relativistic Xα value from Ref. [60]bIn parenthesis are non-relativistic values obtained by setting the velocity of light

to 108 a.u.cNumeric non-relativistic HF value from Ref. [29]

charge distributions, however the calculations within this dissertation

are done employing the Gaussian charge distribution [61,62]. The dif-

ference in the total energies and in orbital energies of atoms obtained

with the use of relativistic four-component formalism with different

nuclear models was studied by Visscher and Dyall [61] and was found

to be sufficiently small, 10−3%, even for an element as heavy as fer-

mium (Z = 100). The advantage of Gaussian nuclear model is that the

analytic formulae for the molecular integrals are easily available.

The use of Eqns. 2.42 and 2.44 for the calculation of Mossbauer

isomer shift has the clear advantage that relativistic and electron cor-

relation effects can be conveniently included in the calculation. The

derivatives of the electronic energy with respect to the nuclear radius


(r.m.s nuclear radius) can be done numerically, which bypasses the

need of computing the relaxed density matrix, enabling to employ

even those computational schemes for which the relaxed density ma-

trix is not available.

Chapter 3

Testing DFT methods in the

calculation of MIS

Synopsis

In this chapter we test a number of density functional methods in the cal-

culation of Mossbauer isomer shifts in a series of 57Fe compounds. The ob-

served trends will help us to discriminate the functionals in terms of their

performance and to employ them for the calibration of α. The influence of

the choice of density functional on the results of calculations reveal that the

hybrid density functionals, especially BH&HLYP, provide better correlation

with experimental results than pure density functionals. The analysis of basis

set truncation reveals that the addition (or removal) of the tightmost primi-

tive functions to a large uncontracted basis set has only a minor influence on

the calculated isomer shift values.

3.1 Introduction

The performance of different quantum chemical methods for the cal-

culation of MIS has to be tested within the linear response formalism

explained in Chapter 2. In the theoretical determination of the isomer

shift we employ a linear relationship between the contact density dif-

ferences in the target compound ρ(a) and in the reference compound

52 3. Testing DFT methods in the calculation of MIS

ρ(s) with the observed values of the isomer shift,

δ = α(ρ(a) − ρ(s)) (3.1)

where α is the calibration constant. If an unreliable method or ba-

sis set is employed, the calibration leads to a dependance of α on the

method/ basis set. For instance, there are many values reported in

the literature varying from -0.15 to -0.37 a30 mm s−1 for α(57Fe). Hence,

it is our hope that we can obtain a more reliable value of α if we em-

ploy the most accurate quantum chemical methods in connection with

the linear response formalism. In order to perform an unbiased test-

ing of the different quantum chemical methods, we do not attempt

parametrisation of α in this chapter, but employ a value of α from

the literature. In the initial study, a series of wave function methods

ranging from the HF method to the CCSD(T) method are tested in

combination with large un-contracted basis sets [58].

However, the use of the advanced wave function methods may be

prohibitively costly for calculations on large biological systems or on

cluster models of solids. The methods based on density functional

theory are more preferable in this respect. Another aspect, the depen-

dence of the results on the basis set truncation was not addressed in

the initial study. Therefore, in the present chapter we would like to

address two issues: i) sensitivity of the results to the choice of the ba-

sis set, and ii) utility of density functional methods for the calculation

of MIS within the new approach.

3.1.Introdu

ction53

Table 3.1: HF calculations of Mossbauer isomer shifts for different iron containing clusters by using rel-ativistic and non-relativistic methods. All shifts are given with respect to [Fe(CN)6]4− (δ = -0.02), see

Refs. [29, 58, 63].

Exptl. Ref.a 24s15p9d3f 23s15p9d3f 22s15p9d3f 21s15p9d3f 20s15p9d3f

1[Fe(H2O)6]2+ 1.41 12 0.89(0.63)b 0.81(0.63) 0.83 (0.63) 0.81(0.63) 0.79(0.63)

2 [FeCl4]2− 0.92 12 0.65(0.35) 0.65(0.35) 0.67(0.37) 0.66(0.37) 0.65(0.37)

3 [Fe(H2O)6]3+ 0.52 12 0.22(0.16) 0.22(0.16) 0.22(0.16) 0.22(0.16) 0.21(0.15)

4 [FeF6]3− 0.50 12 0.26(0.19) 0.26(0.19) 0.26 (0.19) 0.26(0.19) 0.24 (0.19)

6 [FeBr4]1− 0.29 42 0.10(-0.03) 0.08(-0.03) 0.08 (-0.03) 0.04(-0.03) -0.02(-0.03)

7 [FeCl4]1− 0.22 42 -0.02(-0.02) -0.02(-0.02) -0.02(-0.02) -0.02(-0.02) -0.02(-0.02)

8 [Fe(CN)6]3− -0.11 12 -0.20(-0.17) -0.21(-0.17) -0.22(-0.17) -0.20(-0.10) -0.22(-0.17)

9 [Fe(CO)5] -0.12 33 -0.09(-0.07) -0.09(-0.07) -0.09(-0.07) -0.09(-0.07) -0.10(-0.07)

10 [Fe(CO)4]2− -0.12 33 0.07(0.06) 0.05(0.06) 0.05(0.06) 0.03(0.04) 0.03(0.04)

12 [FeO4]2− -0.67 43 -0.99 (-0.77) -0.99(-0.77) -0.99(-0.77) -0.99(-0.77) -0.97 (-0.77)

asources of experimental values.bIn paranthesis, the results of non-relativistic calculations.


For the testing of the wave function based methods within the new

approach [58], the proportionality constant α = -0.1573 a30 mm s−1 de-

termined from the experimental parameters of the 57Fe nuclear tran-

sitions reported in Ref. [6] is used. Note that this value differs by a

factor of more than 2 from the proportionality constants calibrated by

an empirical fit of the calculated electron densities versus the observed

isomer shifts. (see eg. Refs. [29, 31, 63–65]). In the present chapter, we

used this calibration constant α in connection with our new approach

to judge the performance of different density functionals. In the later

chapters of this dissertation, it is our goal to obtain reliable values of119Sn and 57Fe calibration constants α, with the use of accurate wave

function methods and density functionals [66, 67].

3.2 Computational Details

All calculations in this dissertation were carried out using COLOGNE

2005 [68] suite of programs in which the new computational scheme

is implemented. The relativistic calculations were carried out within

the one-electron approximation [69] and using the normalized elimi-

nation of the small component (NESC) [41] method which was imple-

mented according to Ref. [70] (see Section 2.1 for details). The non-

relativistic calculations were carried out by setting high value (108

a.u.) for the velocity of light. Throughout this dissertation closed shell

systems are treated with spin restricted formalism and the open-shell

systems with spin unrestricted formalism.

3.2.Com

putation

alD

etails55

Table 3.2: DFT calculations (PBE) of Mossbauer isomer shifts for different iron containing clusters by usingrelativistic and non-relativistic methods. All shifts are given with respect to [Fe(CN)6]4−.

Exptl. 24s15p9d3f 23s15p9d3f 22s15p9d3f 21s15p9d3f 20s15p9d3f

1 [Fe(H2O)6]2+ 1.41 0.70(0.54)a 0.70(0.54) 0.58(0.54) 0.69(0.54) 0.67(0.54)

2 [FeCl4]2− 0.92 0.37(0.29) 0.37(0.29) 0.38(0.30) 0.38(0.30) 0.38(0.30)

3 [Fe(H2O)6]3+ 0.52 0.34 (0.27) 0.34 (0.27) 0.34(0.27) 0.34(0.27) 0.33 (0.27)

4 [FeF6]3− 0.50 0.34(0.26) 0.34(0.26) 0.33 (0.22) 0.32(0.26) 0.33(0.26)

6 [FeBr4]1− 0.29 n.a.(0.20) n.a.(0.20) n.a.(0.20) n.a.(0.20) 0.14(0.20)

7 [FeCl4]1− 0.22 0.21(0.17) 0.19(0.17) 0.21(0.17) 0.20(0.17) 0.20(0.17)

8 [Fe(CN)6]3− -0.11 -0.06(-0.05) -0.06(-0.05) -0.04(-0.04) 0.01(-0.05) -0.06(-0.05)

9 [Fe(CO)5] -0.12 -0.02(-0.03) -0.02(-0.03) -0.02(-0.03) -0.02(-0.03) -0.02(-0.02)

10 [Fe(CO)4]2− -0.12 0.07(0.06) 0.07(0.06) 0.07(0.06) 0.06(0.05) 0.06(0.06)

12 [FeO4]2− -0.67 -0.40(-0.31) -0.41(-0.31) -0.40(-0.31) -0.41(-0.31) -0.39(-0.31)

aIn paranthesis, the results of non-relativistic calculations.


When calculating the isomer shifts, the effective electron density

inside the nucleus was first calculated using Eqn.2.44 (see Section 2.2.2

for details). Then the isomer shifts were calculated from Eqn. 3.1. The

Density Functional Theory methods used here are PBE, BPW91, BLYP,

B3LYP and BH&HLYP. The basis sets employed will be specified in

the following section.

3.3 Results and discussion

In the present chapter, the MIS calculations are carried out for the fol-

lowing series of iron clusters: [Fe(H2O)6]2+, [FeCl4]2−, [Fe(H2O)6]3+,

[FeF6]3−, [FeI4]1−, [FeBr4]−1, [FeCl4]

−1, [Fe(CN)6]3−, [Fe(CO)5], [Fe-

(CO)4]2−, [Fe(CN)5NO]2− and [FeO4]2−. The geometries were taken

from the compilation in Refs. [63] and from Ref. [71] (Fe(CO)5). The

MIS values for these compounds range from large positive value of

+1.48 mm/s for [Fe(H2O)6]2+ to -0.69 mm/s for [FeO4]2−. The sources

of experimental values for each of these clusters are reported in Table

3.1.

First we would like to address the question of the basis set depen-

dence of the MIS calculated according to the linear response formal-

ism.

3.3.1 Influence of basis sets on isomer shift values

In the preliminary study, reported in Ref. [58], the large un-contracted

basis sets for iron complexes were employed. These basis sets were

constructed by augmenting the standard (20s12p9d) Fe basis set of

Partridge [72] with four tight primitive s-type Gaussian functions and

with a set of polarization functions taken from the TZVpp basis set

3.3. Results and discussion 57

of Ahlrichs and May [73]. The so-obtained (24s15p9d3f) basis set for

iron was combined with the un-contracted aug-cc-pVDZ basis set of

Dunning [74] on other atoms with the only exception of iodine for

which the 6-311G* basis set [75] was used.

In the present chapter, we carry out calculations with this basis

set (denoted further on as basis set A+) for the above mentioned set

of compounds (see Tables 3.1 and 3.2) to make a connection to the

previous work with this method [58]. In the calculations carried out

at the Hartree-Fock and PBE density functional levels of theory, we

study the effect of truncation of the tight primitive basis functions on

the theoretical MISs. The results of the calculations are reported in

Table 3.1 (Hartree-Fock) and in Table 3.2 (PBE density functional).

Analysis of the HF results suggests that the MIS calculated with

Eqns. 2.44 & 3.1 are not very sensitive to the truncation of tight prim-

itive basis functions. In most cases, there is only a modest variation

(ca. 10 %) in the MIS obtained at the relativistic level of theory. The

only marked exception is the iron bromide cluster (see entry 6 in the

Tables), for which the use of truncated basis sets leads to a certain de-

terioration of the results obtained with the inclusion of relativity. The

non-relativistic HF results obtained with Eqn. 3.1 do not show any

noticeable dependence on the truncation of the basis set.

The same trends, a weak dependence of the relativistic results and

an independence of the non-relativistic results on the basis set trunca-

tion, is observed in the density functional calculations (see Table 3.2).

In most cases, the variation in the calculated MIS is of the order of

10 % or less. The use of basis set augmented with tight s-type primi-

tives led, in the case of relativistic density functional calculations for

iron bromide, to serious convergence problems. The source of these

problems is most likely in the use of the numeric quadratures inap-


Table 3.3: Comparison of DFT, HF, and MP2 results for basis sets A+ and B(see text for details of basis sets).

A a B b

Exptl. PBE HF MP2 PBE HF MP2

1 [Fe(H2O)6]2+ 1.41 0.70 0.89 0.96 0.62 0.72 0.82

2 [FeCl4]2− 0.92 0.38 0.66 0.56 0.35 0.36 0.49

3 [Fe(H2O)6]3+ 0.52 0.34 0.22 0.37 0.32 0.21 0.32

4 [FeF6]3− 0.50 0.34 0.26 0.41 0.31 0.25 0.35

5 [FeI4]1− 0.31c n.a. 0.04 0.30 0.48 0.02 0.19

6 [FeBr4]1− 0.29 n.a. 0.10 0.32 0.21 0.00 0.19

7 [FeCl4]1− 0.22 0.21 -0.02 0.18 0.20 0.002 0.17

8 [Fe(CN)6]3− -0.11 -0.06 -0.20 0.05 -0.05 -0.18 0.04

9 [Fe(CO)5] -0.12 -0.02 -0.09 0.00 -0.02 -0.07 0.02

10 [Fe(CO)4]2− -0.12 0.06 0.03 0.05 0.07 0.01 -0.26

11 [Fe(CN)5NO]2− -0.12d n.a. -0.23 -0.17 n.a. -0.16 0.02

12 [FeO4]2− -0.67 -0.40 -0.99 -0.31 -0.32 -0.82 0.04

aby using basis set A+.bby using basis set B.cRef.42.dRef.5.

propriate for relativistic calculations with tight functions. In the non-

relativistic calculations, no convergence problems were observed with

the use of the very tight functions in the basis sets.

The results reported in Tables 3.1 and 3.2 suggest that the con-

verged theoretical results can be obtained with the use of (21s15p9d3f)


iron basis set which is augmented with only one tight primitive func-

tion. This basis set (denoted further on as basis set A) will be used

in further study of the accuracy of different density functional meth-

ods. This basis set is however too big to be used in practical calcula-

tions on large molecular systems. The use of standard basis sets, such

as the (14s11p6d3f)/[8s7p4d1f] basis set of Wachters [76] (Fe) and 6-

31+G* [77] Pople’s basis set on non-metal atoms, is a common practice

in the calculations on large metal complexes [31]. This basis set is de-

noted further on as basis set B. In the current chapter we employed

the DZVP basis set [78] on iodine.

Table 3.3 reports the results of MIS calculations carried out at the

HF, MP2 and DFT (PBE) levels of theory with the use of the basis set B.

These results are compared with the results of the calculations carried

out with the use of the large A+ basis set. The results obtained with the

density functional method are surprisingly stable with respect to the

replacement of the large un-contracted basis set A+ with a standard

contracted basis B. The greatest difference (ca. 20 %) is for the iron

tetroxide cluster which may require extended basis set for the correct

description of the ligand back donation effects. Note however that the

effect of relativity is less visible with the use of a small basis set B. This

is understandable, because this basis set does not contain tight basis

functions needed to describe the relativistic contraction of the electron

density.

603.Testin

gD

FT

methods

inthe

calculation

ofM

ISTable 3.4: Calculation of Mossbauer isomer shifts(mm/s) for different iron containing clusters by usingbasis set A (see text for details of basis set). All shifts are given with respect to [Fe(CN)6]4−.

Exptl. PBE BPW91 BLYP B3LYP BH&HLYP HF

1 [Fe(H2O)6]2+ 1.41 0.69(0.54)a 0.69(0.54) 0.67(0.52) 0.71(0.56) 0.77(0.60) 0.81(0.63)

2 [FeCl4]2− 0.92 0.38(0.30) 0.38(0.30) 0.37(0.29) 0.39(0.31) 0.43(0.34) 0.66(0.37)

3 [Fe(H2O)6]3+ 0.52 0.34(0.27) 0.32(0.26) 0.32(0.25) 0.28(0.22) 0.24(0.18) 0.22(0.16)

4 [FeF6]3− 0.50 0.32(0.26) 0.32(0.26) 0.32(0.25) 0.30(0.23) 0.28(0.21) 0.26(0.19)

5 [FeI4]1− 0.31 n.a. n.a. n.a. n.a. n.a. 0.09(0.07)

6 [FeBr4]1− 0.29 n.a. n.a. n.a. n.a n.a. 0.04(-0.03)

7 [FeCl4]1− 0.22 0.20(0.17) 0.20(0.34) 0.39(0.33) 0.32(0.29) 0.24(0.23) -0.02(-0.02)

8 [Fe(CN)6]3− -0.11 0.01(-0.05) -0.06(-0.05) 0.02(-0.05) 0.00(-0.06) -0.04(-0.02) -0.20(-0.10)

9 [Fe(CO)5] -0.12 -0.02(-0.03) -0.02(-0.03) -0.03(-0.03) -0.04(-0.03) -0.05(-0.04) -0.09(-0.07)

10 [Fe(CO)4]2− -0.12 0.06(0.05) 0.06(0.05) 0.06(0.05) 0.05(0.04) 0.01(0.02) 0.03(0.04)

11 [Fe(CN)5NO]2− -0.12 n.a. n.a. n.a. -0.20(-0.16) -0.28(-0.22) -0.19(-0.10)

12 [FeO4]2− -0.67 -0.41(-0.31) -0.41(-0.32) -0.41(-0.32) -0.51(-0.40) -0.63(-0.49) -0.99(-0.77)

MAEb 0.26(0.30) 0.25(0.31) 0.28(0.31) 0.24(0.28) 0.21(0.26) 0.23(0.26)

Slope 0.48(0.38) 0.49(0.38) 0.46(0.37) 0.53(0.43) 0.61(0.47) 0.78(0.56)

Intercept 0.04(0.02) 0.03(0.04) 0.06(0.04) 0.00(0.00) -0.05(-0.03) -0.14(-0.11)

aIn paranthesis, the results of non-relativistic calculations.bMean Absolute Error of the method.


The HF results in Table 3.3 are even less sensitive to the replace-

ment of the un-contracted basis set with the standard contracted basis

set. However, the MP2 results show much greater sensitivity to the

basis set truncation. The greatest discrepancy is observed for the iron

tetroxide cluster (entry 12), which indicates that the proper descrip-

tion of its electronic structure can not be achieved with the use of the

small basis set at the MP2 level.

To summarize this subsection, the calculations of MIS carried out

with the use of different basis sets suggest that reasonable results can

be obtained at the Hartree-Fock or at the density functional levels of

theory with the use of standard contracted basis sets. However, the

use of small basis sets in connection with the MP2 method leads to a

considerable deterioration of the results for certain compounds. This

suggests that extended basis sets, such as the basis sets A or A+, need

to be used in connection with MP2.

3.3.2 Isomer shift variation with different theoretical

levels

The results reported in Table 3.3 show that the inclusion of electron

correlation via density functionals has a noticeable effect on the cal-

culated MIS. In almost all cases, there is an improvement in the cal-

culated isomer shifts as compared to the HF values. Noticeable im-

provement is obtained for iron halides, cyanides and oxide clusters.

The magnitude of the improvement is comparable with the improve-

ment brought about by MP2 for the large basis set A. For the standard

contracted basis set B, the PBE results are noticeably better than the

MP2 results with one exception of the FeII aqua complex.

In this subsection, we undertake a study of the dependence of the


calculated MIS on the choice of density functional employed. For this

study, we select several popular density functionals: PBE, BPW91,

BLYP, B3LYP, and BH&HLYP. In this selection of functionals, there

are two series which characterize i) the dependence of the results on

specific parametrization of a pure exchange-correlation density func-

tional (series PBE, BPW91, BLYP) and ii) the dependence of the results

on the use of varying fraction of the HF exchange in a hybrid HF/DFT

functional (series BLYP, B3LYP, BH&HLYP). In our opinion, this selec-

tion of functionals enables one to make a reasonable judgement on the

performance of different types of density functionals.

The criteria employed to judge the performance of density func-

tionals in the MIS calculations with Eqns. 2.44 & 3.1 are as follows:

a) The mean absolute error which is a characteristic commonly em-

ployed to judge the overall performance of computational schemes.

b) The slope and the intercept of a least squares linear fit of the exper-

imental vs. calculated isomer shifts, Eqn. 3.2.

δexp = αδcalc + b (3.2)

The parameters α and b characterize the correlation of the calculated

MIS with the experimental values (slope of the linear fit, α) and the

systematic error in the calculated MIS (the intercept of the linear fit,

b).

The results of the density functional and HF calculations are sum-

marized in Tables 3.4 (basis set A) and in Table 3.5 (basis set B). No-

tably the choice of the parametrization of a pure density functional

has a negligible effect on the calculated MIS regardless whether large

(A) or small (B) basis set is employed.

3.3.Resu

ltsan

ddiscu

ssion63

Table 3.5: Calculation of Mossbauer isomer shifts(mm/s) for different iron containing clusters by usingbasis set B (see text for details of basis set). All shifts are given with respect to [Fe(CN)6]4−.

Exptl. PBE BPW91 BLYP B3LYP BH&HLYP HF

1 [Fe(H2O)6]2+ 1.41 0.62(0.53)a 0.62(0.57) 0.60(0.55) 0.61(0.55) 0.65(0.59) 0.72(0.65)

2 [FeCl4]2− 0.92 0.35(0.29) 0.35(0.32) 0.34(0.31) 0.33(0.30) 0.34(0.32) 0.36(0.30)

3 [Fe(H2O)6]3+ 0.52 0.32(0.26) 0.31(0.28) 0.30(0.28) 0.24(0.21) 0.23(0.17) 0.21(0.17)

4 [FeF6]3− 0.50 0.31(0.25) 0.31(0.28) 0.30(0.28) 0.26(0.22) 0.23(0.19) 0.25(0.20)

5 [FeI4]1− 0.31 0.48(0.47) 0.48(0.47) 0.47(0.48) 0.16(0.16) 0.11(0.10) 0.02(0.05)

6 [FeBr4]1− 0.29 0.21(0.19) 0.23(0.23) 0.22(0.21) 0.15(0.14) 0.07(0.07) 0.00(0.00)

7 [FeCl4]1− 0.22 0.20(0.16) 0.20(0.18) 0.19(0.18) 0.13(0.11) 0.06(0.05) 0.00(-0.01)

8 [Fe(CN)6]3− -0.11 -0.05(-0.08) -0.05(-0.05) -0.05(-0.05) -0.09(-0.09) -0.13(-0.13) -0.18(-0.18)

9 [Fe(CO)5] -0.12 -0.02(-0.05) -0.02(-0.02) -0.02(-0.02) -0.05(-0.05) -0.07(-0.07) -0.07(-0.06)

10 [Fe(CO)4]2− -0.12 0.07(0.06) 0.07(0.06) 0.07(0.06) 0.03(0.02) -0.01(-0.01) 0.01(0.01)

11 [Fe(CN)5NO]2− -0.12 n.a. n.a. n.a. -0.16(-0.10) -0.10(-0.10) -0.16(-0.15)

12 [FeO4]2− -0.67 -0.32(-0.34) -0.32(-0.31) -0.33(-0.31) -0.44(-0.42) -0.55(-0.52) -0.82(-0.77)

MAEb 0.25(0.27) 0.25(0.26) 0.25(0.27) 0.24(0.25) 0.23(0.25) 0.25(0.26)

Slope 0.43(0.39) 0.43(0.40) 0.42(0.39) 0.47(0.43) 0.52(0.48) 0.63(0.57)

Intercept 0.07(0.05) 0.07(0.05) 0.07(0.07) -0.02(-0.02) -0.06(-0.07) -0.13(-0.13)

aIn paranthesis, the results of non-relativistic calculations.bMean Absolute Error of the method.


The mean absolute error, and the parameters of the linear fit re-

main nearly the same for different pure density functionals. This ob-

servation suggests that it is sufficient to study the effect of hybridiza-

tion with varying fraction of the HF exchange in one series of hybrid

density functionals only.

Inclusion of the Hartee-Fock exchange in hybrid functionals leads

to a certain increase in the systematic error as given by the intercept

of the linear fit for both basis sets, A and B. At the same time, mixing

in more HF exchange leads to an improved correlation of the calcu-

lated values with the experiment. The slope of the linear fit increases

steadily as the fraction of the HF exchange increases. This effect is

observed for both basis sets, however, the bigger basis set A provides

better overall correlation with the experimental results. The hybrid

functional BH&HLYP gives reasonable correlation with experiment

and less systematic error compared to HF, for both basis sets A and B.

The improvement brought about by the inclusion of HF exchange

in hybrid functionals warrants some discussion. In our opinion, the

most plausible explanation for the inferior performance of pure den-

sity functionals is the incorrect behaviour of the potential generated

by such a functional near the nucleus. It is known that the gradient

corrected functionals yield the Kohn-Sham potential which is diver-

gent at the nuclear position [79]. Therefore, mixing in the HF exchange

(the HF potential remains finite at the nucleus) cures partially this de-

ficiency of pure density functionals and leads to improved results for

the properties which critically depend on the electron distribution in

the vicinity of the nucleus.

3.3.Resu

ltsan

ddiscu

ssion65

Table 3.6: Comparison of Mossbauer isomer shifts (mm/s) from this work (using basis set B) with that ofRef. [31], recalculated according to the current method.

Exptl. HFa BH&HLYPa B3LYPa B3LYPb BPW91a BPW91b

1 [Fe(H2O)6]2+ 1.41 0.72 0.65 0.61 0.55 0.62 0.53

2 [FeCl4]2− 0.92 0.36 0.34 0.33 0.39 0.35 0.38

3 [Fe(H2O)6]3+ 0.52 0.21 0.23 0.24 0.18 0.31 0.21

4 [FeF6]3− 0.50 0.25 0.23 0.26 0.24 0.31 0.27

6 [FeBr4]1− 0.29 0.00 0.07 0.15 0.16 0.23 0.20

7 [FeCl4]1− 0.22 0.02 0.06 0.13 0.15 0.24 0.19

8 [Fe(CN)6]3− -0.11 -0.18 -0.13 -0.09 -0.07 -0.05 -0.05

9 [Fe(CO)5] -0.12 -0.07 -0.07 -0.05 -0.05 -0.02 -0.05

10 [Fe(CO)4]2− -0.12 0.01 -0.01 0.03 -0.08 0.07 -0.06

11 [Fe(CN)5NO]2− -0.12 -0.16 -0.10 -0.16 -0.09 n.a. -0.06

12 [FeO4]2− -0.67 -0.82 -0.55 -0.44 -0.40 -0.32 -0.31

MAEc 0.25 0.23 0.24 0.24 0.25 0.24

Slope 0.63 0.52 0.47 0.45 0.43 0.41

Intercept -0.13 -0.06 -0.02 -0.02 0.07 0.01

aThis work.bRef. [31].cMean Absolute Error of the method.


With the use of both basis sets, A and B, the difference between

relativistic and non-relativistic results is clearly visible. From all the

parameters employed for the data analysis, it is evident that the inclu-

sion of relativity leads to improved results as compared to the experi-

ment. The difference between relativistic and non-relativistic results is

more pronounced for the large basis set A than for the small basis set

B. This is understandable, because the basis set B was optimized and

contracted in the non-relativistic HF calculations and does not possess

sufficient flexibility to accommodate changes in the electron distribu-

tion due to the inclusion of relativity. This underlines the necessity

to develop compact basis sets adapted for the relativistic calculations.

Nevertheless, the overall agreement with the experiment is acceptable

for both hybrid functionals, B3LYP and BH&HLYP. Although the HF

method provides the best correlation with the experimental results

(slope of 0.78 and 0.63 with the basis sets A and B, respectively), the

systematic error increases in the HF calculations as compared to pure

or hybrid density functionals. In the overall assessment, the use of hy-

brid density functionals with increased fraction of the HF exchange,

such as the BH&HLYP functional, can be recommended for the calcu-

lation of MIS in iron complexes.

The computational procedure employed in the present chapter em-

ploys the proportionality constant α in Eqn. 3.1 which was obtained

from the experimentally measured parameters of nuclear transitions

in 57Fe. Therefore, its comparison with the results of the standard

calculations of the MIS, where this constant is treated as an empiri-

cal parameter and is fitted against the experimental data, may be not

straightforward. However, with the use of the contact densities pub-

lished in Ref. [31], it is possible to calculate the MISs with the use of

Eqn. 3.1 and the non-empirical constant α. The so-obtained isomer

3.4. Conclusion 67

shifts are compared in Table 3.6 with the shifts calculated with the use

of Eqns. 2.44 & 3.1 in the present work. From this comparison, it is

obvious that the standard approach does not have any numerical ad-

vantage before the method used in the present article, if the fitting pro-

cedure is excluded. In this sense, the method employed in this chapter

helps to make an unbiased judgement on the performance of the com-

putational schemes used to calculate the MIS. Because it was not the

purpose of the present work to obtain empirically adjusted propor-

tionality constant in Eqn. 3.1, the use of a set of 12 molecules seems to

be acceptable for making a reasonable judgement on the performance

of different computational schemes. The use of the fitting procedure

within the standard approach to MIS, although leads to improved nu-

merical results, does not let to see the true accuracy of a selected quan-

tum chemical method and may result in an unrealistic parameters of

nuclear γ- transitions as obtained from the fitted proportionality con-

stant α in Eqn. 3.1. It is our goal to avoid the empirical fitting and to

find out computational schemes capable of yielding accurate results

from first principles.

3.4 Conclusion

In the present chapter, the linear response approach for the theoreti-

cal calculation of Mossbauer isomer shift is applied to a series of iron

complexes within the context of density functional theory. The com-

putational schemes employed include both hybrid and pure density

functionals as well as the HF method. Before the performance of den-

sity functional methods was addressed, the dependence of the quality

of the calculated MIS on the size of the basis set was studied.


The investigation of the effect of the basis set truncation within

the context of the new approach reveals that the MISs are not very

sensitive to the removal/addition of the tightest primitive functions

from/to large uncontracted basis set. Therefore, for obtaining con-

verged theoretical values, it is sufficient to employ only one tight s-

type primitive function added to the standard uncontracted basis set

(see the basis set A). With the use of the small contracted basis set

(see the basis set B), an acceptable accuracy in the calculated MIS is

obtained for the HF and density functional methods. The use of the

small basis set, such as the basis set B, is therefore a reasonable com-

promise between accuracy and complexity of the calculation.

The investigation of the performance of different density function-

als reveals that the pure density functional methods provide poorer

correlation of the calculated MIS with the experimental values. Irre-

spective of the size of the basis set employed, the hybrid functionals

provide consistently better description of the MIS. Correlation of the

calculated MIS with the experimental values improves with the in-

creasing fraction of the HF exchange, however at a price of somewhat

greater systematic error. In the overall assessment, the hybrid func-

tionals with greater fraction of the HF exchange such as the BH&HLYP

functional, produce better description of the MIS in iron complexes.

The performance of the density functional methods and the ef-

fect of basis set truncation on the calculated isomer shifts evaluated

in the present chapter. However, even within the combination of hy-

brid functionals with greater fraction of the HF exchange such as the

BH&HLYP functional and the large basis set (A), the overall agree-

ment of the calculated isomer shifts with experimental shifts are not

entirely satisfactory (see the mean absolute errors in Table 3.4). It

seems therefore that the use of a more accurate value of the calibra-

3.4. Conclusion 69

tion constant α from the empirical fit of the calculated contact densi-

ties with the experimental shifts, should improve this agreement. The

determination of such a reliable calibration constant requires the use

of highly accurate ab initio quantum chemical calculations.

Generally, the density functional methods demonstrate somewhat

inferior numeric accuracy as compared to the ab initio wavefunction

methods. Therefore, the calibration of α should be done preferably

with the ab initio wave function methods when possible. Having ob-

tained such a method independent calibration constant, one can apply

this value in practical calculations without the necessity to calibrate it

everywhere. The contents of Chapters 4 and 5 presents the parametri-

sation of such a reliable, universally applicable calibration constant α

for 119Sn and 57Fe within the scope of the linear response formalism.

Chapter 4

Calibration of 119Sn isomer shift

Synopsis

In the present chapter, the importance of the effects of relativity and electron

correlation for accurate description of the isomer shift is demonstrated on

the basis of the HF, MP2 and of the CCSD calculations on a series of iron

ions. The isomer shift for the 23.87 keV M1 resonant transition in the 119Sn

nucleus is calibrated with the help of ab initio calculations, the calibration

constant α(119Sn) obtained from MP2 calculations (αMP2(119Sn) = (0.091 ±

0.002)a0−3 mm/s) is in good agreement with the previously obtained values.

The approach used in the calibration is applied to study the 119Sn isomer shift

in CaSnO3 perovskite under pressure. Comparison with the experimental re-

sults for the pressure range 0 GPa to 36 GPa shows that the linear response

approach is capable of describing tiny variations of isomer shift with reason-

able accuracy.

4.1 Introduction

The use of an empirical calibration constant α for the improvement of

the correlation between experiment and theory is suggested in Chap-

ter 3. In the present chapter, the linear response approach is applied

72 4. Calibration of 119Sn isomer shift

to the calibration of Mossbauer isomer shift of 119Sn nucleus. Tin com-

pounds have been extensively studied with the use of the Mossbauer

spectroscopy and there exists a plethora of the experimental data on

the 119Sn isomer shift in various chemical environments [80–82].

In a number of previous studies, calibration constants α(119Sn) rang-

ing from 0.17 a0−3 mm/s [83] to 0.037 a0

−3 mm/s [84] have been de-

duced from the experimental and theoretical data. The most recent

theoretical value of the α(119Sn) constant is (0.092 ± 0.002)a0−3 mm/s

as obtained in periodic density functional calculations of Svane et al.

[82]. Hence accurate theoretical simulation of the experimental data

over a wide range of chemical environments are used to obtain a the-

oretical value of the calibration constant α, which is compared with

the previously obtained values. In this chapter, a special emphasis is

given on the use of systematically improvable ab initio wave function

methods and on the role of electron correlation effects for obtaining

accurate theoretical results.

An independent test of the obtained calibration constant α(119Sn)

is carried out for the theoretical simulations of the pressure depen-

dence of the Mossbauer isomer shift in CaSnO3 perovskite. It is our

hope that the calibration of the 119Sn isomer shift on a representative

set of compounds carried out with the use of accurate computational

methods will help to improve the accuracy of interpretation of the ex-

perimental measurements.

The importance of including proper account of relativity and elec-

tron correlation in the calculations is demonstrated in a series of atomic

calculations, for various oxidation states of iron atom.

4.2. Computational Details 73


The isomer shift of the Mossbauer spectrum is commonly related to

the so-called contact density [5, 6, 12, 29], via Eqn. 3.1 (δ = α (ρ(a) -

ρ(s))), in which α is the calibration constant which depends on the pa-

rameters of the nuclear γ-transition. However, the parameters of the

nuclear γ-transition, such as the variation of the nuclear charge radius,

are not known experimentally with sufficiently high accuracy, such

that the direct determination of α is currently not possible. Presently,

the most popular method of determination of the calibration constant

is based on comparison of the theoretically estimated contact densities

with the experimentally observed isomer shifts [5, 6, 12, 29, 30, 85]. In

this calibration procedure, one uses a linear regression equation,

δ = αρ+ b (4.1)

where the parameters α and b are determined from the least squares

fit. The electron contact densities near the nucleus are calculated ac-

cording to Eqn. 2.44 (see Section 2.2.2 for details).

The calculations in this chapter are carried out both at HF and MP2

level. 21s15p11d2f basis set of Dyall [86] is used for Sn and for all other

elements the augmented correlation consistent double-zeta (aug-cc-

pVDZ) basis set of Dunning [74] is used. All basis sets are used in the

uncontracted form.

In the current and following chapters, we employ embedded clus-

ter approach to calculate the local electronic structure of solids. Within

this approach, a cluster of atoms representing a structural unit of the

crystalline solid is immersed in the Madelung potential of the rest

of the crystal. The Madelung potential is modeled by a large array


of the point charges placed at the appropriate crystallographic posi-

tions. The magnitudes of the charges are determined from the natural

bond order (NBO) analysis [87] of the respective cluster wave function

calculated at the HF level. The co-ordinates of atoms in the clusters

and of the point charges representing the Madelung field were ob-

tained from the crystal structures generated with VESTA [88]. From

our preliminary calculations we concluded that the inclusion of the

Madelung potential of the crystal has relatively minor effect on the

calculated contact densities (Eqn. 2.44) which is consistent with the

conclusions of other works [29, 63]. However, because the clusters

modeling the above solids are negatively charged, the embedding po-

tential was added to compensate for the extra negative cluster charge.

The Sn compounds used in the present investigation are SnF4, SnO2,

CaSnO3, BaSnO3, SnCl4, SnS2, SnBr4, SnSe2, SnI4, SnO, SnS, SnSe and

SnCl2. The isomer shift of these compounds ranges from -0.36 mm/s

(SnF4) to +4.06 mm/s (SnCl2) (see Table 4.2). The sources of experi-

mental values of each of these compounds are cited in Table 4.1. The

crystal structures and the cluster geometries considered in the present

work are explained below. The cluster geometries are represented in

Figure 4.1.

SnF4 [89] has a body-centered tetragonal structure. The considered

cluster is [SnF6]2−, where the tin atom is surrounded by distorted oc-

tahedron of fluorine atoms (see Figure 4.1 (a)). The atomic positions

are in the 14/mmm space group.

SnO2 [90] has a rutile, tetragonal structure (P42/mnm). The clus-

ter on which calculations are done is [SnO6]8−, where the tin atom is

surrounded by a distorted octahedron of oxygen atoms (see Figure 4.1

(b)).

CaSnO3 [91] and BaSnO3 [91] have cubic structures (Pm3m). The


(a) (b) (c) M = Ca/Ba

(d) (e) X = Cl/Br

(f) E = S/Se (g) (h)

(i) E = S/Se (j)

Figure 4.1: The cluster geometries used in the calculations, (a) [SnF6]2−

in SnF4 (b) [SnO6]8− in SnO2 (c) [SnO6]

8−.8M2+, M = Ca/Ba in

CaSnO3/BaSnO3 respectively (d) [SnO6]8−.8Ca2+ in CaSnO3 perovskite (e)

[SnX4], X = Cl/Br in SnCl4/SnBr4 respectively (f) [SnE6]8−, E = S/Se in

SnS2/SnSe2 respectively (g) [SnI4] in SnI4 (h) [Sn4O14]20− in SnO (i) [SnE3]

4−,

E = S/Se in SnS/SnSe respectively (j) [SnCl7]5− in SnCl2.


tin atoms are in a cubic environment where the Ca/Ba ions lie in the

cube vertices and O atoms lie in the middle of all cube edges. The clus-

ter considered is [SnO6]8−, with the eight Ca2+/Ba2+ ions around the

cluster modeled by the respective Stuttgart Effective Core Potentials

(ECPs) [132] (see Figure 4.1 (c)).

CaSnO3 considered in the Section III B, has an orthorhombically-

distorted perovskite structure [93](Pnma). It consists of corner linked

SnO6 octahedra with the Ca ions in the nine-fold oxygen coordination.

The cluster considered is [SnO6]8− with eight Ca2+ ions around the

cluster (which is modeled by Stuttgart ECPs). The tin atom is in an

environment of oxygen octahedron (see Figure 4.1 (d)).

SnCl4 [95] and SnBr4 [96] have monoclinic structures (P21/c) with

four molecules per unit cell. The molecular clusters of SnCl4 and SnBr4

with distorted tetrahedral geometry are considered for the calcula-

tions (see Figure 4.1 (e)).

SnS2 [97] and SnSe2 [97] have a layered structure(P3m1). The sul-

fide or selenide anions form a hexagonal close packed arrangement

while the tin cations fill alternating layers of octahedral sites. The

crystal consists of X-M-X sandwiches (where M is Sn and X is S in SnS2

and Se in SnSe2). The unit cell contains two molecules. The considered

clusters are [SnS6]8− and [SnSe6]

8−, where the coordination around the

Sn atom is a distorted octahedron (see Figure 4.1 (f)).

SnI4 [98] have cubic lattice (Pa3) with eight molecules packing loosely

in the unit cell. The cluster considered is SnI4 with distorted tetrahe-

dral geometry (see Figure 4.1 (g)).

SnO [99] is in tetragonal structure (P4/nmm) with four tin atoms

in the unit cell. The chemically meaningful structural unit of this crys-

tal includes four closely packed SnO4 tetragonal pyramidal blocks

pairwise connected via bridged oxygen atoms. For this crystal, we


(a) (b)

Figure 4.2: Contact density difference ∆ρ(Fen+) = ρ(Fen+) − ρ(Fe0) as afunction of metal charge n. (a) All calculations are carried out for the iso-

lated Fen+ atoms in the respective high-spin ground states with the use of

the non-relativistic and relativistically corrected CCSD(T) method (see the

legend). (b) All calculations are carried out for the isolated Fen+ atoms in

the respective high-spin ground states with the use of the relativistically cor-

rected HF, MP2 and CCSD(T) methods (see the legend).

included all the four tin atoms in the unit cell into the representa-

tive cluster, which resulted in the [Sn4O14]20− cluster embedded in the

Madelung field of the rest of the crystal (see Figure 4.1 (h)).

SnS [100] and SnSe [100] have an orthorhombic structure (Pnma).

The clusters considered for these solids are [SnS3]4− and [SnSe3]

4− in

which the tin atom has three pyramidal S/Se neighbors, two located

in the plane of the layer and one at a short distance normal to this

plane (see Figure 4.1 (i)).

SnCl2 [101] has an orthorhombic structure (Pnam) with four SnCl2

units per unit cell. All atoms in the unit cell lie in two planes parallel


to (001). The considered cluster is [SnCl7]5−, in which the tin atom is

surrounded with three chlorine atoms at distances of 2.68, 3.21, 3.30 A

in the same plane, two chlorine atoms at distances of 2.78 A and two

chlorine atoms at a distance of 3.05 A(see Figure 4.1 (j)).

4.3 Results and discussion

In this section, we present the results of the theoretical determination

of the 119Sn calibration constant αwhich is obtained from the linear re-

lationship (Eqn. 4.1) between the theoretically calculated contact den-

sities with the experimental isomer shifts in a series of tin compounds.

The calibration parameters obtained from the linear regression analy-

sis will be used to calculate the 119Sn isomer shifts, which will be com-

pared with the available experimental values. As an independent test

of the quality of obtained calibration parameters a calculation of the

isomer shift in CaSnO3 perovskite under pressure will be undertaken.

Before we start with the calculations on cluster models, we would

like to demonstrate the importance of inclusion of the effects of rela-

tivity and electron correlation into the calculation of the isomer shift.

Traditionally, for elements as light as 57Fe, an approach based on

the use of the contact densities obtained from the non-relativistic cal-

culations with the point-like nucleus is employed [5, 6, 12]. Because

relativity strongly modifies the wave function and the electron den-

sity in the vicinity of the nucleus, the use of the non-relativistic con-

tact density ρ = ρ(0) (where ρ(0) denotes the electron density at the

nuclear position) may lead to considerable errors even for a light ele-

ment. This is illustrated in Figure 4.2 (a), where the contact density dif-

ferences ∆ρ(Fen+) = ρ(Fen+) − ρ(Fe0) calculated within the linear re-


sponse formalism [58] for a series of iron ions are shown. In this figure,

the contact density differences obtained from the relativistically cor-

rected CCSD(T) atomic calculations are compared with those obtained

in the non-relativistic CCSD(T) calculations for the ground states of

respective iron ions. When taking the density differences, the non-

relativistic contact densities are scaled with a factor ρrel(Fe0)/ρnr(Fe

0)

which agrees with the commonly adopted prescription of scaling of

the non-relativistic contact densities [12]. It can be seen that relativity

has a noticeable effect on the density differences and, in some cases

(e.g., Fe+ and Fe4+), may account for up to 50% of the total density

variation. Because the isomer shift is a relative quantity and depends

on the density differences rather than on the total contact densities, the

importance of the proper account of relativity is obvious from Figure

4.2 (a).

Another factor which can modify the density distribution is the

electron correlation. The importance of electron correlation for the

contact densities is illustrated in Figure 4.2 (b), where the density dif-

ferences ∆ρ(Fen+) = ρ(Fen+) − ρ(Fe0) obtained in the relativistically

corrected Hartree-Fock, MP2 and CCSD(T) calculations for a series of

iron ions are shown. From comparison of the HF and CCSD(T) den-

sity differences, it is obvious that electron correlation makes a notice-

able contribution to the overall density variation, especially for highly

charged ions. It is noteworthy that the results of the MP2 calcula-

tions closely match the CCSD(T) results. This observation suggests

that the MP2 method can be used in the calculations on cluster mod-

els of solids instead of the CCSD(T) method, which should lead to a

considerable saving of the computation time.


Table 4.1: The electron contact densities of 119Sn clusters (a large constant of182800 a0

−3 has been subtracted from all the values) along with the structural

references.

Space group Ref. HF MP2

SnF4 14/mmm [89] 13.32 23.84

SnO2 P42/mnm [90] 17.50 29.68

CaSnO3 Pm3m [91] 15.25 27.92

BaSnO3 Pm3m [91] 16.46 26.51

SnCl4 P21/c [95] 27.50 35.19

SnS2 P3m1 [97] 27.37 40.60

SnBr4 P21/c [96] 30.94 40.39

SnSe2 P3m1 [97] 30.37 43.66

SnI4 Pa3 [98] 35.28 45.17

SnO P4/nmm [99] 44.00 55.50

SnS Pnma [100] 57.03 64.23

SnSe Pnma [100] 58.65 65.28

SnCl2 Pnam [101] 67.43 72.14

4.3.1 Calibration of the 119Sn isomer shift

The electron contact densities calculated using Eqn. 2.44 are reported

in Table 4.1 along with structural references. The results of linear re-

gression analysis of the theoretical contact densities versus the exper-

imental isomer shifts are presented in Figure 4.3. Besides the contact

densities obtained in the present work with the use of the relativisti-

cally corrected HF and MP2 methods, the contact densities obtained


Figure 4.3: Calculated electron contact density (in a0−3) vs experimental iso-

mer shifts (in mm/s) for Sn clusters using HF, MP2 and DFT(LDA) [82] re-

spectively.

by Svane et al. [82] as an average of the total electronic densities ob-

tained in periodic density functional calculations with the local den-

sity approximation (LDA) functional inside a sphere of nuclear charge

radius, are also used in the linear regression analysis and are reported

in Figure 4.3.

The calibration parameters obtained from the linear regression in

Figure 4.3 are αHF(119Sn) = (0.081 ± 0.002)a0−3 mm/s, bHF = (-14807.91

± 467.11) mm/s and αMP2(119Sn) = (0.091 ± 0.002)a0

−3 mm/s, bMP2

= (-16584.65 ± 378.40) mm/s respectively. The αMP2 value is in good

agreement with the calibration constant αLDA(119Sn) = (0.092± 0.002)a0−3

mm/s, obtained from the periodic LDA calculations [82]. Given that

both methods, MP2 and LDA, include the electron correlation, good

agreement between the results of different sets of calculations indi-

cates the importance of inclusion of the correlation effects into the iso-

mer shift calculations. The results of the statistical analysis of the data

reported in Table 4.1 and in Figure 4.3 support this conclusion. The

MP2 method shows a much lower standard deviation in the linear


regression analysis and an improved r2 correlation coefficient as com-

pared to the HF values, σMP2 = 0.114 mm/s vs. σHF = 0.158 mm/s and

r2MP2 = 0.994 vs. r2HF = 0.989.

It is noteworthy that a systematic improvement of the results of the

theoretical calculations can be achieved with the use of the methods

which take the electron correlation into account. Indeed, the electron

correlation results in a noticeable contraction of the electron density

towards the nucleus, which is evident from the contact densities re-

ported in Table 4.1 and from the literature data [102]. Besides that, the

inclusion of electron correlation results in an improved description

of the bond covalency in metal complexes, which plays an important

role for a reliable determination of the contact densities as has been

demonstrated by Sadoc et al [85]. In this respect, it is rather surprising

that the LDA periodic calculations [82] yield somewhat inferior sta-

tistical correlation with the experimental data than the MP2 embed-

ded cluster calculation, σMP2 = 0.114 mm/s vs. σLDA = 0.163 mm/s.

This suggests that with the use of the wave function based methods

one achieves more efficient system-specific account of the important

electron correlation effects. Although the comparison has been made

against the LDA calculations, judging from the results of density func-

tional calculations of Mossbauer isomer shifts [31, 63, 103] the use of

the gradient-corrected functionals should not lead to a marked im-

provement of the correlation with experiment. The use of the hybrid

functionals, however, may lead to an improved correlation with ex-

periment and may bring the accuracy of density functional calcula-

tions closer to the wave function methods.

The calibration parameters α(119Sn) and b(119Sn) obtained with the

HF and MP2 methods were used to calculate the 119Sn isomer shifts

using Eqn. 4.1. The resulting 119Sn isomer shifts are compared in Table


Table 4.2: Experimental isomer shifts and isomer shifts calculated accordingto Eqn. 4.1 using the contact densities given in Table 4.1 and calibration

constants (from linear fits in Figure 4.3).

δexp Ref. HF MP2 LDAa

SnF4 -0.36 [104] -0.23 -0.37 -0.28

SnO2 0.00 [16] 0.03 -0.01 0.07

CaSnO3 0.00 [16] 0.11 -0.13 -0.28

BaSnO3 0.00 [16] -0.07 0.15 -0.06

SnCl4 0.85 [95] 0.92 0.65 n.a.

SnS2 0.98 [94] 0.91 1.15 1.23

SnBr4 1.13 [105] 1.20 1.13 n.a.

SnSe2 1.36 [17, 106] 1.15 1.42 1.48

SnI4 1.55 [105] 1.55 1.56 n.a.

SnO 2.64 [107] 2.26 2.50 2.63

SnS 3.29 [94, 108–110] 3.31 3.29 3.43

SnSe 3.31 [106, 108] 3.44 3.38 3.21

SnCl2 4.06 [111] 4.50 4.01 3.94

M.A.E.b 0.13 0.08 0.12

σc 0.15 0.11 0.16

M.S.E.d 0.02 0.01 0.01

aCalculated using the densities and calibration constants obtained from the peri-

odic LDA calculations in Ref. 22.bMean Absolute ErrorcStandard deviationdMean Signed Error


Figure 4.4: Calculated isomer shifts (in mm/s) vs experimental isomer shifts(in mm/s) for Sn clusters using HF, MP2 and DFT(LDA) [82] respectively.

4.2 with the experimental values and with the values obtained from

the calibration parameters and contact densities reported in Ref. [82].

The results of the linear regression analysis of the so-obtained theo-

retical isomer shifts versus the experimental values are presented in

Figure 4.4. Again, the MP2 method provides an improved statistical

correlation of the theoretical isomer shift values with the experiment.

It is gratifying that the MP2 method provides better statistical cor-

relation with the experiment than the earlier reported periodic LDA

calculations of Svane et al. [82], σMP2 = 0.112 mm/s vs. σLDA = 0.160

mm/s. The average errors in the calculations also strengthen this ar-

gument, M.A.E.MP2 = 0.08 mm/s vs M.A.E. LDA = 0.12 mm/s (refer

Table 4.2). Taken together with an improvement in the calibration

procedure brought about by the inclusion of the electron correlation

in the MP2 method, this observations provides strong evidence of the

importance of proper description of the electron correlation effects in

the calculation of isomer shift in metal compounds.


4.3.2 Isomer shift variation in CaSnO3 perovskite

The calibration constants obtained in the previous subsection were

employed to calculate the variation of 119Sn isomer shift in CaSnO3

perovskite. Recently, the experimental results for the dependence of

the isomer shift under external pressure have been obtained in Ref.

[112]. The range of variation of the isomer shift is -0.08 mm/s for

the pressure range of 0 GPa to 36 GPa. No phase transitions have

been observed in CaSnO3 perovskite under high pressure, such that

the isomer shift variation is entirely due to the change of the local

geometry around the tin sites [112].

For a theoretical method employed for the calculation of isomer

shift, variation of the isomer shift of CaSnO3 perovskite under pres-

sure represents a stringent test. Indeed, the range of isomer shift in

this case is rather narrow, much narrower than the range of isomer

shifts in different compounds used in the previous subsection. There-

fore, the ability of a theoretical method to reproduce these variations

with a reasonable accuracy should indicate the quality of the compu-

tational approach. In the previous work [112], a value of +0.008 mm/s

was obtained at the pressure of 36 GPa by using traditional approach

based on the MO-LCAO (molecular orbital as a linear combination

of the atomic orbitals) calculations. Besides being an order of magni-

tude too small, this value has the wrong sign which indicates certain

inconsistencies in the theoretical method employed in Ref. [112].

In the present chapter, we employed the unit cell parameters of

CaSnO3 perovskite determined in Ref. [113] for the pressure range

from 0 GPa to 8.5 GPa. The fractional coordinates of atoms in the unit

cell were taken from Ref. [93] at 0 GPa pressure. Because the informa-

tion on the fractional atomic coordinates of CaSnO3 perovskite under


Figure 4.5: Pressure variation (in GPa) vs isomer shift (in mm/s) for CaSnO3

perovskite structure. Experimental values are taken from Ref. [82].

non-zero pressure is not available from the literature, these coordi-

nates were kept fixed in the calculations. The lattice parameters for

the pressure greater than 8.5 GPa were obtained by an extrapolation

of the parameters obtained experimentally for the 0 - 8.5 GPa pressure

range. Although the lack of accurate information on the fractional co-

ordinates and the lattice parameters may lead to certain errors in the

calculations, we adopted this approach in the present work following

Ref. [112], where the unit cell parameters were obtained in a similar

way.

The dependence of the 119Sn isomer shift in CaSnO3 perovskite cal-

culated with the use of the HF and MP2 methods is plotted in Fig. 4.5

along with the experimental curve [112]. For each theoretical method,

the calibration constants α and b (see Eqn. 4.1) obtained in the pre-

vious subsection are employed. The maximum variations of the 119Sn

isomer shift obtained with the use of the HF and MP2 method are -0.14

mm/s and -0.20 mm/s, respectively. Note that the theoretical isomer

shift variations have the same sign as the experimentally measured


ones and that the theoretically obtained curves in Fig. 4.5 follow the

profile of the experimental curve [112].

The discrepancy between the calculated and the experimental vari-

ations of the isomer shift, most likely, can be attributed to the inac-

curacies in the determination of the geometry of the clusters used in

the present calculations. Note that with the use of similarly obtained

local geometries a much lower value of +0.008 mm/s of the isomer

shift variation at 36 GPa was obtained in Ref. [112]. It is also note-

worthy that, in the pressure range 0 - 8.5 GPa, where at least the ex-

perimental lattice parameters are available, a discrepancy between the

calculated and experimental isomer shifts is rather small. The second-

order Doppler shift (SODS) due to the lattice vibration represents yet

another factor which can influence the results of the isomer shift mea-

surements under high pressure [120]. The omission of this contribu-

tion in the calibration of the isomer shift does not affect the quality

of calibration because all experimental data have been obtained un-

der the ambient pressure. Although the SODS makes a negligibly

small contribution at the ambient pressure [120] it may affect the rel-

ative isomer shifts calculated with respect to the ambient pressure,

such as those reported in Figure 4.5. Therefore, the comparison with

the experiment is incomplete and does not allow one to draw a clear

conclusion on the superior performance of one of the methods. Nev-

ertheless, the results reported in Figure 4.5 give us confidence that

the present theoretical scheme is capable of providing qualitative esti-

mates of tiny variations of the isomer shift due to pressure. Because, in

the presence of phase transitions, the isomer shift variation is typically

much greater [114, 115] than that found in Ref. [112], we expect that

the present theoretical method should enable one to describe the vari-

ations of isomer shift in phase transitions with sufficient confidence.


4.4 Conclusion

Reliable theoretical determination of the hyperfine parameters of Mo-

ssbauer active nuclei still remains one of the challenging tasks in com-

putational chemistry. The interpretation of Mossbauer isomer shift

necessitates the determination of a method-independent value of the

calibration constant α, which provides connection between the elec-

tron density in the vicinity of the nucleus and the isomer shift via Eqn.

2.44. In the present chapter, we have undertaken a theoretical study

of the calibration constant α of the 119Sn nucleus using the linear re-

sponse approach.

With the use of linear response approach to the isomer shift the im-

portance of proper description of the effects of relativity and electron

correlation for the theoretical contact densities was demonstrated for

a series of iron ions. It was shown that relativity and electron correla-

tion may account for up to ca. 50% of the contact density differences

which are important for the isomer shift.

The calibration constant αMP2(119Sn) = (0.091 ± 0.002)a0

−3 mm/s

obtained in the present chapter with the use of relativistically cor-

rected MP2 calculations is in excellent agreement with the previously

obtained value. This gives us confidence that a method-independent

value of the calibration constant can indeed be determined in theoret-

ical calculations. The improvement in the linear regression analysis

of the calculated contact densities with respect to the observed isomer

shifts using the MP2 method is noticeable from the statistical correla-

tion (see Figure 4.3). Besides that, the description of the isomer shifts

calculated with the correlated method (MP2) are in a much better

agreement with the experimental values than the HF results (see Table

4.2 and Figure 4.4). This finding implies that a systematic improve-

4.4. Conclusion 89

ment in modeling of Mossbauer parameters can be achieved with the

use of systematically improvable ab initio methods. It is expected that

even better agreement with the experiment can be achieved with the

use of more sophisticated methods of quantum chemistry, such as the

coupled cluster method (CCSD(T)) or the multi-reference many-body

perturbation theory method (CASPT2).

An independent verification of the quality of the 119Sn calibration

constants obtained in this work is achieved in the calculation of vari-

ation of the 119Sn isomer shift in CaSnO3 perovskite under pressure.

Reasonable agreement of the theoretical values of the isomer shift in

the pressure range 0-36 GPa with the experiment has been reached.

However, more precise information on the variation of the lattice pa-

rameters and atomic fractional coordinates is necessary to provide a

more reliable description of isomer shifts. Nevertheless, the ability of

the linear response approach to correctly describe very tiny variations

of the isomer shift resulting from the change of local geometry under

pressure is encouraging. It is our hope that with the use of the current

theoretical approach one can achieve a reliable interpretation of the

experimental Mossbauer spectra in situations where simple models

based on intuitive arguments may fail.

The improvement in the MIS calculations brought about by the

use of systematically improvable ab initio methods demands the test-

ing of the double hybrid density functional B2-PLYP developed by S.

Grimme [55]. This is carried for the parametrization of α(57Fe), along

with the HF and MP2 methods, within the linear response approach

and described in Chapter 5.

Chapter 5

Calibration of 57Fe isomer shift

Synopsis

In the present chapter, the recently developed double hybrid density func-

tional B2-PLYP along with the HF and MP2 methods are used to obtain a re-

liable value of the the calibration constant α(57Fe). The calibration constant

obtained within B2-PLYP functional (α(57Fe) = -0.306 ± 0.009 mm/s) is in

good agreement with the experimentally estimated constant of −0.31± 0.04

a30 mm s−1 and can be recommended for the theoretical modeling of 57Fe iso-

mer shifts.

5.1 Introduction

The calibration procedure based on Eqn. 4.1 (see Chapter 4) is fol-

lowed in the present chapter to obtain a reliable value of the 57Fe cal-

ibration constant. In many works, a very good linear correlation be-

tween δ and ρ were obtained. However, the value of α could vary in

a wide range of values depending on the method of calculation and

a set of experimental isomer shifts employed in the calibration. Thus,

for iron 57Fe a ”consensus” value of −0.267±0.115 a30 mm s−1 was sug-

gested by Oldfield from a compilation of the published α(57Fe) con-

stants [116]. Wide error bars in this value imply that there is no clear

92 5. Calibration of 57Fe isomer shift

convergence of the estimated calibration constants to a single value.

Besides theoretical calculations, the contact densities can be evalu-

ated from the experimentally measured life times of electron capture

by atomic nuclei. Thus, a value of −0.31 ± 0.04 a30 mm s−1 was de-

rived by Ladriere et al. [117] from the life-time measurements of elec-

tron capture by the 52Fe nucleus. Because the electron capture rate is

proportional to the contact density ρ, the α(57Fe) calibration constant

could be obtained from a comparison of the measured life times with

the 57Fe isomer shifts in the same compounds.

A substantial difference between this value and the theoretically

estimated ”consensus” value of α(57Fe) suggests that certain short-

comings of theoretical calculations need to be improved. In particu-

lar, the effects of relativity and electron correlation need to be accu-

rately taken into account in the theoretical calculations. We have al-

ready seen the importance of these effects in Chapter 4, which strongly

modify the electron distribution in the vicinity of the nucleus thus af-

fecting the theoretical contact density (see Figure 4.2). The ability of

the linear response approach to include relativity and electron corre-

lation effectively, lead to a reliable calibration of the isomer shift of119Sn with the use of high-level ab initio calculations (see Chapter 4).

The improvement in the calibration brought in with the use of highly

correlated methods in Chapter 4, demands the testing of the recently

developed double hybrid density functional B2-PLYP [55] for the iso-

mer shift parametrizations.

In the present chapter, we apply the linear response formalism to

the calibration of 57Fe isomer shift on the basis of ab initio calcula-

tions carried out with the wave function theory methods, such as the

Hartree-Fock method, the second-order Møller-Plesset (MP2) pertur-

bation theory method, as well as with the density functional method


(a) (b) (c) X = F/Br/I/Cl

(d) (e) (f)

(g) (h)

Figure 5.1: The cluster geometries used in the calculations, (a) [FeF6]4−.8K1+

in KFeF3 (b) [Fe(H2O)6]2+/3+ (c) [FeX6], X = F/Br/I/Cl in FeF2, FeBr2, FeI2

and FeCl2 respectively (d) Fe(C5H5) (e) [Fe(CN)6].8K+ in K4Fe(CN)6 (f)

[Fe(CN)6].8K+ in K3Fe(CN)6 (g) Fe(CO)5 (h) [FeO4]2−.9K+ in K2FeO4.


B2-PLYP. The major purpose of this chapter is to apply the most accu-

rate computational schemes in combination with large un-contracted

basis sets to obtain a reliable value of the α(57Fe) calibration constant.

A representative number of iron-containing solid compounds (crys-

talline solids and matrix isolated molecules) is employed in the cali-

bration with the use of the embedded cluster approach.


The electron contact densities near the nucleus are calculated accord-

ing to Eqn. (2.44) and the calibration constant α is obtained from Eqn.

(4.1) by linear regression of the calculated densities against the exper-

imental isomer shifts.

The calculations are carried out using HF, MP2 and double hy-

brid density functional (B2-PLYP) level. The basis set denoted as A+

in Chapter 3 is used for iron and the correlation consistent double-

zeta (cc-pVDZ) basis set of Dunning [74] is used for the neighbouring

atoms. All basis sets are used in un-contracted form.

The iron compounds used in this work are FeF2, KFeF3, [Fe(H2O)6]2+,

FeBr2, FeCl2, FeI2, [Fe(H2O)6]3+, FeF3, FeCp2, K4Fe(CN)6, K3Fe(CN)6,

Fe(CO)5, K2FeO4. The geometries of the cluster models of these com-

pounds are explained in the following.

KFeF3 has a perovskite structure (Pm3m) [118]. The considered

cluster is [FeF6]4− with eight K+ ions (see Figure 5.1 (a)) around the

cluster modeled by the respective Stuttgart effective core potentials

(ECPs) [132]. The rest of the crystal (up to 5 unit cells in each crystal-

lographic direction) is modeled with the point charges.

In [Fe(H2O)6]3+, the iron atom is octahedrally coordinated by six


Table 5.1: 57Fe electron contact densities according to Eqn. 2.44 calculatedwithin relativistically corrected HF, MP2 and B2-PLYP methods.

HFa MP2a B2-PLYPb

FeF2 86.33 86.00 8.59

KFeF3 86.41 86.03 8.72

Fe(H2O)62+ 86.56 86.05 8.68

FeBr2 87.31 87.16 9.74

FeCl2 87.24 87.06 9.52

FeI2 88.03 88.14 10.51

Fe(H2O)63+ 90.35 89.92 11.75

FeF3 89.98 89.30 11.76

FeCp2 90.12 89.91 11.50

K4Fe(CN)6 91.69 92.07 13.55

K3Fe(CN)6 92.79 91.49 13.72

Fe(CO)5 92.39 92.19 13.98

K2FeO4 93.78 93.15 15.47

aa large constant of 14800 a0−3 has been subtracted from all the values.

ba large constant of 15000 a0−3 has been subtracted from all the values.

water molecules. In its coordination sphere, the H atoms of water

molecule are arranged in the Th symmetry. In [Fe(H2O)6]2+ (Ci sym-

metry) a distortion in the octahedral complex occurs due to the Jahn-

Teller effect. The geometries of the [Fe(H2O)6]2+ and [Fe(H2O)6]

3+

complexes (see Figure 5.1 (b)) are taken from Sadoc et al. [85]. For

the aqua complexes, the effect of the surrounding water was modeled

with the Polarizable Continuum Model (PCM) [119].


FeF2 has a body-centered tetragonal structure (P4/mnm) [120]. FeBr2

[120] and FeI2 [121] have rhombohedral structures (P3m1). FeCl2 has a

monoclinic structure (R3m) [120] and FeF3 has a rhombohedral struc-

ture (R3c) [122]. An octahedrally coordinated [FeX6] complex is con-

sidered for these compounds where X is F in FeF2/FeF3, Br in FeBr2,

Cl in FeCl2 and I in FeI2 (see Figure 5.1 (c)).

FeCp2 has two centrosymmetric molecules of Fe(C5H5) in the unit

cell and the iron atoms lie in the symmetry centers (D5h symmetry)

(see Figure 5.1 (d)). The average Fe-Cp distances are 1.66 A, C-C dis-

tances 1.40 A, C-H distances 1.104 A and the H-C-Cp bond angles

of 125.98◦ [123]. Because the available crystal structure [124] is not

suitable for setting up the Madelung field, the effects of the environ-

ment is taken into account via the PCM approach in the argon matrix.

Fe(CO)5 is a trigonal bipyramid [125] for which the effect of argon

matrix is modeled via the PCM method (see Figure 5.1 (g)).

K4Fe(CN)6 (C2/c) [126] (see Figure 5.1 (e)) and K3Fe(CN)6 (P21/c)

(see Figure 5.1 (f)) [127] have monoclinic structures. The clusters con-

sidered are the octahedrally coordinated [Fe(CN)6] complex with eight

K+ ions around the cluster which are modeled by ECPs.

K2FeO4 has an orthorhombic structure (Pnam) [128]. The consid-

ered cluster is [FeO4]2− with nine K+ions around it. The oxygen atoms

are in tetrahedral coordination around the iron (see Figure 5.1 (h)).

5.3 Results and Discussion

In this section we apply the linear response approach to the calibration

of 57Fe Mossbauer isomer shift in a series of compounds described in

the previous section.

5.3. Results and Discussion 97

Figure 5.2: Linear regression of the contact densities calculated with the rela-tivistically corrected HF, MP2 and B2-PLYP methods for a series of iron com-

pounds against the experimental isomer shifts. The calibration constant α,

the Pearson correlation coefficient r2 and the standard deviation σ are given

for each method.

Table 5.1 reports the contact densities (Eqn. (2.44)) calculated with

the use of the relativistically corrected HF, MP2, and B2-PLYP meth-

ods. These densities are used in the linear regression analysis with

Eqn. (4.1) the results of which are plotted in Figure 5.2. The param-

eters of linear fit (Eqn. (4.1)) are used to calculate the isomer shifts

which are reported in Table 5.2. The values of the calibration con-

stant obtained with different methods, αHF(57Fe) = −0.265 ± 0.009 a30

mm s−1, αMP2(57Fe) = −0.271±0.013 a30 mm s−1, and αB2−PLYP(

57Fe) =

−0.306±0.009 a30 mm s−1, approach the experimental estimate of −0.31±

0.04 a30 mm s−1 [117] with increasing accuracy of the electron correla-

tion description in the sequence HF < MP2 < B2-PLYP.

The B2-PLYP calculations provide the most reliable value of the

calibration constant αB2−PLYP(57Fe) = −0.306 ± 0.009 a30 mm s−1 as

evidenced by the smallest mean absolute deviation (0.047 a30 mm s−1)

and the standard deviation (0.072 a30 mm s−1) of the calculated isomer


shifts from the experimental values presented in Table 5.2.

Table 5.2: 57Fe isomer shifts (in mm/s) obtained from a linear regres-sion, Eqn. 4.1, of the the ρcalc. (Table 5.1) against the δexp.. For each

method, the parameters of Eqn. (4.1) as well as the Pearson correla-

tion coefficient r2 and the standard deviation σ are given.The relative

isomer shifts ∆δx = δx - δref (in mm/s) calculated with respect to dif-

ferent compounds are given in the lower part of the table.

δexp HF MP2 B2-PLYP

FeF2 1.467 1.445 1.461 1.460

KFeF3 1.440 1.424 1.453 1.420

Fe(H2O)62+ 1.390 1.384 1.447 1.432

FeBr2 1.120 1.185 1.146 1.108

FeCl2 1.092 1.204 1.173 1.175

FeI2 1.044 0.995 0.880 0.872

Fe(H2O)63+ 0.500 0.381 0.398 0.493

FeF3 0.480 0.479 0.566 0.489

FeCp2 0.460 0.442 0.400 0.569

K4Fe(CN)6 -0.020 0.026 -0.186 -0.058

K3Fe(CN)6 -0.130 -0.265 -0.028 -0.110

Fe(CO)5 -0.140 -0.159 -0.218 -0.190

K2FeO4 -0.690 -0.527 -0.478 -0.646

α -0.265 ± 0.009 -0.271 ± 0.013 -0.306 ± 0.009

b 3941.6 ± 140.2 4038.6 ± 194.0 4595.4 ± 136.3

r2 0.986 0.975 0.990

M.A.E. 0.059 0.089 0.047

σ 0.086 0.116 0.072

continued on next page


Table 5.2 – continued from previous page

HF MP2 B2-PLYP

w.r.t. K4Fe(CN)6

M.A.E. 0.0836 0.1793 0.0658

σ 0.0998 0.2140 0.0833

w.r.t. KFeF3

M.A.E. 0.0618 0.0949 0.0551

σ 0.0881 0.1166 0.0753

w.r.t. FeF3

M.A.E. 0.0639 0.1165 0.0523

σ 0.0863 0.1486 0.0729

w.r.t. K2FeO4

M.A.E. 0.1765 0.2292 0.0649

σ 0.1970 0.2575 0.0865

To test the reliability of the obtained calibration constants we have

undertaken calculations of the relative isomer shifts ∆δx = δx−δref by

choosing different reference compounds. The results of these calcula-

tions are collected in lower part of Table 5.2, where the relative shifts

∆δx calculated with respect to K4Fe(CN)6, KFeF3, FeF3 and K2FeO4

are shown. It is seen that the use of the B2-PLYP method and the

αB2−PLYP(57Fe) calibration constant leads to the best correlation with

the experimental values of the relative shifts irrespective of the refer-

ence compound selected. Thus we can conclude that the αB2−PLYP(57Fe)

value of −0.306±0.009 a30 mm s−1 is the most reliable theoretical value

of the calibration constant obtained in this work. It is gratifying that

this value is in a very good agreement with the experimental esti-

mate of −0.31± 0.04 a30 mm s−1 for the 57Fe calibration constant [117].

Note that the αB2−PLYP constant obtained in this work is in a good


agreement with the value of -0.311 a30 mm s−1 obtained by Romelt, Ye

and Neese [63] from the B2-PLYP calculations employing the quasi-

relativistic ZORA Hamiltonian, however it is somewhat lower in the

absolute value than the constant -0.336 a30 mm s−1 obtained in the non-

relativistic B2-PLYP calculations.

5.4 Conclusions

With the use of relativistically corrected HF, MP2, and B2-PLYP meth-

ods the values of the calibration constant α(57Fe) were obtained from a

linear regression of the contact densities calculated for a large number

of iron compounds modelled within the embedded cluster approach

against the experimental isomer shift values. The most reliable theo-

retical value of −0.306 ± 0.009 a30 mm s−1 was obtained with the use

of the B2-PLYP method which provides the best statistical correlation

(r2 = 0.990 and σ = 0.072 a30 mm s−1) with the experimental data.

The reliability of the obtained calibration constant was also verified

in the calculation of the relative isomer shifts ∆δx = δx − δref using

different iron compounds as a reference. In all cases, the B2-PLYP

method with αB2−PLYP(57Fe) = −0.306 ± 0.009 a30 mm s−1 yielded the

least deviations from the experimental data. It is noteworthy that this

theoretical value is in a very good agreement with the experimental

estimate of −0.31 ± 0.04 a30 mm s−1 [117]. This observation suggests

that the α(57Fe) value obtained in this work can be used as a universal

constant in theoretical modelling of the Mossbauer isomer shift.

The reliable parametrization of the 57Fe calibration constant α in

the present chapter allow us to use the obtained calibration parame-

ters for the computation of isomer shifts within various applications,

5.4. Conclusions 101

and are described in the following chapters. Chapter 6 details the in-

vestigation on the presence of different iron sites in a Prussian blue

analogue, RbMn[Fe(CN)6].H2O. The understanding of the chemical

structure variation around the 57Fe coordination sphere for the re-

cently discovered iron-based superconductors is described in Chapter

7.

Chapter 6

Study of the presence of different iron

sites in RbMn[Fe(CN)6]·H2O

Synopsis

The 57Fe electron contact densities and the corresponding Mossbauer isomer

shifts of RbMn[Fe(CN)6]·H2O are calculated using the calibration parame-

ters obtained in Chapter 5. The theoretical analysis of possible origin of the

two slightly different signals in the Mossbauer spectra observed at 293 K and

50 K (Vertelman et al. 2008 Chem. Mater. 20 1236-38) is carried out. The

measured differences of 0.083 - 0.105 mm/s between the high-temperature

and low-temperature phases can be attributed to the different oxidation states

of iron: the iron is in FeIII oxidation state in the high temperature phase and

FeII state in the low temperature phase. The smaller isomer shift differences

of 0.012 - 0.034 mm/s occurring in both high temperature and low temper-

ature structures can be attributed to different distributions of the RbI ions

within the unit cell of RbMn[Fe(CN)6]·H2O.

6.1 Introduction

In the present chapter, using the results obtained in Chapter 5 we

undertake a theoretical analysis of 57Fe Mossbauer isomer shifts in

104 6. Study of the presence of different iron sites in RbMn[Fe(CN)6]·H2O

a Prussian blue analogue RbMn[Fe(CN)6]·H2O. Prussian blue ana-

logues are compounds with the general molecular formula AxMa[Mb-

(CN)6]·zH2O (where, A = alkali cation and Ma/Mb = metal ion), which

attract considerable interest due to their peculiar magneto-optical prop-

erties [129]. Thus, RbMn[Fe(CN)6]·H2O shows light and temperature

induced switching of magnetization [130]. Upon cooling from 300K

to around 150K, RbMn[Fe(CN)6]·H2O undergoes transition to a state

with higher molar magnetization. A reversed transition to a low mag-

netization state is observed upon heating to ca. 250K. It has been hy-

pothesized that the switching of magnetization occurs due to electron

transfer from the MnII cations to the FeIII ions (and reversed transi-

tion) as a result of structural phase transition upon cooling/heating

the samples [131].

Table 6.1: 57Fe Mossbauer isomer shifts observed in RbMn[Fe(CN)6]·H2O atdifferent temperatures [131].

T, K δ, mm/s

50 -0.033(4)a -0.067(5)

293 -0.138(2) -0.150(2)

aNumber in the parentheses shows the experimental uncertainty in the last digit.

The Mossbauer spectra of the high-temperature (HT) and the low-

temperature (LT) structures were recorded whereby it was observed

that, in both structures, two signals with the isomer shifts reported in

Table 6.1 were present [131]. The presence of two different FeIII sites

in the HT phase of RbMn[Fe(CN)6]·H2O with the shifts δ = −0.138(2)

mm/s and δ = −0.150(2) mm/s was interpreted as a consequence


of different distributions of the Rb cations over the interstitial sites 4c

and 4d (see Figure 6.2) in the spacegroup F 43m of the crystal.

Figure 6.1: Crystal structure ofRbMn[Fe(CN)6]·H2O.

Although it was not possible to obtain reliable crystal structures of

the LT phase of RbMn[Fe(CN)6]·H2O, it is clear that iron is present in

the form of low-spin FeII ions octahedrally coordinated by the cyano

ligands. Similar to the HT phase, a disorder in the occupation of the

Rb sites around the FeII ions leads to emergence of two different iron

sites as can be seen from the isomer shifts reported in Table 6.1.

Figure 6.2: Two different iron sites in RbMn[Fe(CN)6]·H2O originating fromdifferent distribution of the Rb ions, the first cluster shown is referred as Type

I and second as Type II respectively [131].

To obtain insight into the origin of different iron sites in the HT and

LT structures of RbMn[Fe(CN)6]·H2O we undertake quantum chem-

ical calculations of 57Fe Mossbauer isomer shifts in cluster models of

these sites. More specifically, our goal is to elucidate whether the ob-

served isomer shifts originate from variations in the charge state of


iron ions (FeII vs. FeIII isomer shifts) and from differences in the co-

ordination sphere of iron ions due to different distribution of the Rb1

ions (see Figure 6.2).


The calculations are carried out using HF, MP2, CCSD and double

hybrid density functional (B2-PLYP [55]) level. The A+ basis set dis-

cussed in Chapter 3 is used for iron and 6-31G* [77] basis set is used for

carbon and nitrogen. The Rb1 core is modeled by the Stuttgart effec-

tive core potentials (ECPs) [132], whereas MnII/III cores are modeled

by the representative ECPs of CaII and AlIII, respectively.

The ∆δs are obtained using B2-PLYP functional by substituting the

obtained contact densities (Eqn. 2.44) in Eqn. (4.1), where αB2−PLYP(57Fe)

= -0.306 ± 0.009 mm/s (see Chapter 5 for details).

Table 6.2: 57Fe electron contact densities and isomer shift differences ob-tained at different theory levels.

FeIII FeII ∆ρ ∆δ

HF 14940.12 14938.17 -1.95 0.60

MP2 14939.82 14940.16 0.35 -0.11

B2-PLYP 15140.89 15140.18 -0.71 0.22

CCSD 14940.70 14940.38 -0.52 0.16

The geometries of clusters shown in Figure 6.2 are obtained from

crystal structure of the HT phase reported in Ref. [131]. The HT phase


has a unit cell which belongs to cubic space group F43m with lattice

constant a = 10.521 A. The iron is in octahedral coordination with six

cyano ligands bridging Fe and Mn. The disorder in the distribution

of RbI ions is modeled by selecting two possible distribution patterns

of RbI around iron atoms as shown in Figure 6.2. The Fe-C and Mn-N

distances are 1.929 A and 2.205 A respectively. The same geometry of

clusters is used for the LT phase calculations.


In this section, the influence of different factors, both electronic and

geometric, on the 57Fe contact densities of RbMn[Fe(CN)6]·H2O crys-

tal are presented and discussed in detail. The main goal is to deter-

mine the origin of the different Fe sites observed in the HT and LT

phases (see Table. 6.1) [131].

6.3.1 Influence of the different oxidation states and co-

ordination sites

In RbMn[Fe(CN)6]·H2O, iron can present either in the FeII or FeIII oxi-

dation states depending on the charge transfer between iron and man-

ganese. In this chapter, we have carried out calculations on both oxi-

dation states of iron incorporated in clusters described in the previous

section. In both oxidation states, iron is assumed to be present in a

low-spin state: spin 0 in FeII clusters and spin 1/2 in FeIII clusters.

Table 6.2 gives the 57Fe electron contact densities and isomer shifts

obtained at different levels of theory which are also represented pic-

torially in Figure 6.3 (type I coordination sphere shown in Figure 6.2


Figure 6.3: 57Fe isomer shift differences obtained at differenttheory levels in type I clusters (see Fig. 6.2).

is assumed in these calculations). From the earlier theoretical works

on K4Fe(CN)6 and K3Fe(CN)6, ∆δ (FeII - FeIII) is expected to be pos-

itive [29]. The Hartree-Fock, B2-PLYP and CCSD yield positive ∆δ

values while MP2 yields a negative value (see Table 6.2 and Figure

6.3). From Table 6.2 and Fig. 6.3, it is obvious that the ∆δ values ob-

tained with the use of the methods including electron correlation are

in much better agreement with the experimental observations than the

HF value (∆δCCSD = 0.16 mm/s and ∆δB2−PLYP = 0.22 mm/s vs. ∆δHF =

0.60 mm/s). The values obtained from the CCSD and B2-PLYP calcu-

lations are very close to one another. Therefore B2-PLYP is employed

in the further calculations as a cost effective alternative to the coupled

cluster method.

Table 6.3 gives the contact densities and isomer shifts on iron in FeII

and FeIII oxidation states, for two different distribution patterns of Rb

around Fe (see Fig. 6.2). In both types of Rb distribution patterns,


Table 6.3: 57Fe electron contact densities and isomer shifts at different oxida-tion states incorporated in Type I and Type II clusters (see Figure 6.2).

Type I Type II ∆ρ ∆δ

FeIII 15140.89 15140.72 -0.18 0.05

FeII 15140.18 15139.98 -0.20 0.06

∆ρ -0.71 -0.74

∆δ 0.22 0.23

change in the oxidation state of iron leads to a large variation of the

isomer shift, of the order of ∆δ (FeII - FeIII) ∼ 0.2 mm/s. These large

variations in δ of iron sites with iron in FeII and FeIII oxidation states

agrees with the experimentally observed variations of the isomer shift

in HT and LT phases. Thus, it can be concluded that iron in the HT

phase is present predominantly in the form of FeIII, whereas in the LT

phase it is in the form of FeII.

Changing the distribution pattern of Rb1 ions (see Figure 6.2) leads

to much smaller variations of the isomer shift for both oxidation states

of iron (see Table 6.3). The ∆δ values reported in the last column of

Table 6.3 are consistent with the isomer shift differences of ∆δHT =

0.012 mm/s and ∆δLT = 0.034 mm/s observed for different iron sites

within the HT phase and the LT phase respectively (see Table 6.1).

Thus, these differences can be interpreted as originating from differ-

ences in the Rb1 ions distribution patterns around the FeIII sites in the

HT phase and around the FeII sites in the LT phase.


Figure 6.4: Difference ∆δ(FeII − FeIII) = δ(FeII(RFe−C)) -δ(FeIII(RFe−C = 1.929A)) as a function of the Fe - C distance

in the FeII coordination sites.

6.3.2 Influence of the bond length variations

In the calculations reported in the previous subsection, it was assumed

that the geometry of the coordination sphere of iron ions is the same

in the HT and in the LT phases. In the HT phase, crystal structure of

which is known experimentally, the FeIII ions are surrounded by six

cyano ligands at a distance RFe−C = 1.929 A. Because changing the oxi-

dation state of iron to FeII should lead to contraction of the Fe - C bond

length [29, 63], we have studied the effect of the bond length contrac-

tion on the contact densities and isomer shifts. The results of calcula-

tions of the isomer shift on FeII sites (type I Rb1 coordination sphere

is assumed) with respect to the isomer shift on the FeIII sites at the ex-

perimental geometry, ∆δ(FeIIRFe−C− FeIIIexp)) as a function of the Fe - C

distance (RFe−C) are presented in Figure 6.4. It is obvious that the bond

6.4. Conclusion 111

length contraction in FeII coordination sites leads to a decrease in the

isomer shift difference. Therefore, the value of ∆δ(FeII-FeIII) = 0.22

mm/s obtained for the type I coordination at the HT experimental ge-

ometry should be corrected to a lower value (e. g. ∆δ ≈ 0.09 mm/s

for FeII - C bond length of 1.89 A), thus improving the agreement with

the experiment (see Table 6.1) [131].

6.4 Conclusion

The origin of the different iron sites observed experimentally in the57Fe Mossbauer spectra of Prussian blue analogue RbMn[Fe(CN)6]·H2O

is analyzed with the use of the calibration constant α (57Fe) obtained

in Chapter 5. The calculations have been carried out at different the-

oretical levels varying from the relativistically corrected Hartree-Fock

method to the coupled cluster method (CCSD) and to the double hy-

brid density functional method (B2-PLYP) of Grimme [55].

The theoretically obtained 57Fe isomer shift values enable one to in-

terpret the experimentally observed ∆δHT−LT isomer shift differences

(ca. 0.1 mm/s, see Table 6.1) between the HT phase and the LT phase

of the RbMn[Fe(CN)6]·H2O compound as originating from the change

of the oxidation state of iron from FeIII to FeII. Two different iron sites

observed within the HT and LT phases separately, which lead to ∆δ

values of the order of 0.01 - 0.03 mm/s (see Table 6.1), can be inter-

preted as originating from different distribution patterns of the Rb1

ions around the iron sites as shown in Figure 6.2. The theoretical cal-

culations yield the ∆δ values of the order of 0.05 mm/s for the two

types of Rb1 coordination thus corroborating the analysis of the ex-

perimental measurements presented by Vertelman et al. [131].

Chapter 7

Theoretical investigation

of hyperfine parameters of iron-based

superconductors

Synopsis

The aim of the present chapter is to study the the influence of the structural

variation on the isomer shift and magnetic hyperfine coupling constants in

iron based superconductors. The calculations based on the linear response

approach indicate a decrease of the electron contact density near the 57Fe nu-

cleus with the tetragonal-orthorhombic phase transition in iron pnictides and

chalcogenides. Thus, such a decrease of the contact density and correspond-

ing increase of 57Fe isomer shifts can be used for the identification of the local

chemical environment in these compounds. The isotropic hyperfine coupling

constant Aiso of the studied iron based superconductors did not make no-

ticeable differences with the structural phase transition. However, with the

inclusion of spin-orbit corrections in the calculations the Aiso values make

distinct variation between different types of these compounds. The SO cor-

rections scale up to 30 % of the calculated Aiso values, and is necessary for

the accurate description of magnetic properties in these compounds.

1147. Theoretical investigation of hyperfine parameters of iron-based

superconductors

7.1 Introduction

The discovery of high-temperature superconductivity (HTSC) in iron-

based compounds has initiated a considerable research activity com-

parable or even in excess to the discovery of HTSC in cuprates [22].

Iron-based superconductors belong to pnictide (compounds of group

V elements) [23–25] or chalcogenide (compounds of group VI ele-

ments) class of compounds [26]. One can distinguish several types

of iron- based superconductors: a) the REFeAsO systems with RE =

Rare Earth = La, Nd, Ce, etc (see Figure 7.1(d)). These compounds

are often denoted as ”1111” type and become superconducting upon

doping with fluorine. For these compounds, the maximum critical

temperature Tc = 56 K was achieved for samarium oxyarsenide. b)

The ”122” systems with the formula AFe2As2 (A = Ba, Sr, Ca) (see

Figure 7.1(c)) which have critical temperature comparable to the 1111

compounds, the superconductivity emerges with the appropriate sub-

stitution of bivalent A cations with monovalent alkali metals, K, Cs etc.

c) The 111 compounds with the formula LiFeAs and Tc =18K (see Fig-

ure 7.1(b)). d) The 011 chalcogenides FeSe with Tc ≈ 8.5 K (see Figure

7.1(a)). Most of these systems become superconducting upon doping

or application of external pressure.

These compounds contain structures with layers of edge-sharing

FeX (X = As, Se, etc.) tetrahedra separated by metal ions. The un-

doped 1111 compounds, which are non-superconducting, exhibit struc-

tural, tetragonal-orthorhombic phase transition upon cooling below

90K and a long-range magnetic order that sets in at the temperature

of the structural transition [133]. Other compounds, notably 011 type,

do not show any long-range magnetic order although undergo the

structural transition [26]. It is worth to mention that for LiFeAs, no


evidence of a structural phase transition is observed so far [134]. Al-

though the relationship between the structure, magnetism and super-

conductivity is still unresolved, there are claims that the suppression

of both the structural transition and magnetic order, or only the mag-

netic order, or neither are necessary for superconductivity to emerge

[135].

(a) (b)

(c) (d)

Figure 7.1: The iron-based superconductors, (a)FeSe (011 type) (b) LiFeAs(111 type)(c) BaFe2As2 (122 type) (d) LaFeAsO (1111 type).

All iron based superconductors have a two dimensional (2-D) struc-

ture, which is shown in Figure 7.1. Therefore, at a glance their physical


superconductors

properties are considered to be highly two-dimensional. In resolving

the origin of superconductivity in these iron based compounds, the

knowledge of their local geometry and local electronic structure is ex-

tremely important. The parameters of Mossbauer spectra, such as the

isomer shift, quadrupole splitting and magnetic hyperfine splitting,

carry information on the spin- and oxidation state of the resonating

atom and its local chemical environment. This technique can be used

on amorphous/disordered systems and one can monitor phase transi-

tions, mixed phases, disordered structures etc. The aim of the present

chapter is to find a relation between the local chemical environment

of the iron based superconductors and the Mossbauer parameters, iso-

mer shifts and hyperfine coupling constants. This analysis will give us

an idea on how can these Mossbauer parameters be used to judge the

structure of the iron based superconductors, especially under phase

transitions and upon doping/chemical substitutions.

7.2 Details of calculations

The iron chalcogenides and iron pnictides studied are, FeSe, LiFeAs

(111 type), BaFe2As2 (122 type) and SmOFeAs (1111 type). The fol-

lowing geometries were used in this study.

The iron chalcogenide, FeSe crystallizes in tetragonal (P4/nmm)

structure at 298 K and in orthorhombic structure (Cmma) at 4 K, com-

posing a stack of edge-sharing FeSe4 tetrahedra with no intercala-

tion layer [26]. The considered cluster is [FeSe4]6− (see Figure 7.2(a)).

LiFeAs crystallizes in tetragonal unit cell (P4/nmm) and there is no

structural phase transition with the lowering of temperature. The

cluster considered is [Li4FeAs4]6− (see Figure 7.2(b)).

7.2. Details of calculations 117

The 122 iron pnictide, BaFe2As2 crystallizes in tetragonal (14/mmm)

structure at 298 K where the crystal contains layers of edge-sharing

FeAs4/4 tetrahedra separated only by barium atoms [24]. The phase

transition occurs at 140 K resulting to orthorhombic space group Fmmm.

The considered cluster model is [Ba4FeAs4]2− (see Figure 7.2(c)).

SmOFeAs has a tetragonal structure (P4/nmm) with alternating

layers of Fe-As and Sm-O in which these layers are well separated [23].

These compounds experience a structural phase transition at around

150 K, resulting in the orthorhombic symmetry of the Cmma space

group. The considered cluster model is [Sm4FeAs4O4]6− (see Figure

7.2(d)).

The MIS calculations were carried out using the B2-PLYP func-

tional and HFC (hyperfine coupling constant) calculations using the

B3LYP functional. The 24s12p9d basis set of Partridge with a set of

polarization functions taken from TZVpp basis set of Ahlrichs and

May is used for iron, which is in the uncontracted form. For As/Se

the augmented correlation consistent double-zeta (aug-cc-pVDZ) ba-

sis set of Dunning is used. The LiI and BaII were modeled by the

respective Stuttgart effective core potentials (ECPs). SmIII is modeled

by the AlIII ECPs, having same valence state and nearly identical ionic

radii the use of AlIII helps to avoid the complexity of including f elec-

trons in Sm. All the calculations were done in embedded cluster ap-

proach, which is explained in Chapter 4.

The Mossbauer isomer shift calculations were carried out using the

linear response formalism, as described in the earlier chapters. The

contact density near the nucleus is calculated according to Eqn. 2.44,

and the MIS are calculated using the calibration constant α(57Fe) =

−0.306 ± 0.009 a30 mm s−1 obtained with the use of the B2-PLYP func-

tional in Chapter 5 (using Eqn. 3.1).


superconductors

(a) (b)

(c) (d)

Figure 7.2: The clusters studied, (a)FeSe4 (b) Li4FeAs4 (c) Ba4FeAs4 (d)Sm4FeAs4O4.

The details of the HFC calculations are given below. The part of the

ESR spin Hamiltonian referring to hyperfine coupling is usually writ-

ten as [136],

HhfK = S.AK .IK (7.1)

The isotropic Fermi-contact term AFC, and the anisotropic dipolar cou-

pling Adip, can be calculated using the Breit-Pauli Hamiltonian and a

vector potential corresponding to a point-like magnetic dipole mo-

ment of nucleus K. Within a single-determinantal LCAO approach,

these may be expressed as [136],

AisoK = AFC

K =4π

3α2geγK

1

2 < SZ > µ, ν

∑

P α−βµ,ν < χµ|δ(rK)|χν >

(7.2)

AdipK,µ,ν =

1

2α2gegKγK

1

2 < SZ > µ, ν

∑

P α−βµ,ν < χµ|r

−5K (r2Kδµ,ν

−3rK,νrK,ν)|χν >

(7.3)

7.2. Details of calculations 119

where γK is the gyromagnetic ratio of nucleus K, and ge is the free-

electron g value (2.002319...). <SZ> is the maximum value of the spin

projection and Pµ,να−β is the spin density matrix in the atomic orbital

basis {χµ}. AFC and Adip contribute to the nonrelativistic part of the

HFC tensor. The details on the theory behind HFC calculations can be

obtained from Ref. [136] and the references therein.The dominant SO

corrections to the calculations are included as a second order correc-

tion [136].

The total A tensor is expressed as,

AK = ANRK + ASO

K (7.4)

where ASOK is the spin-orbit correction to A, which may be obtained in

terms of an isotropic pseudocontact (PC) and traceless dipolar (”dip,2”)

term.

ASOK = APC

K + Adip,2K (7.5)

The HFC calculations were done with the use of the MAG-ReSpect

program package in which the coupled perturbed Kohn-Sham algo-

rithm with the inclusion of spin-orbit interaction is implemented [137].

The calculations are done with iron spin S = 1 state.

Generally in physics and mathematics, uniaxiality and asymmetry

are two characteristics of a symmetric second-rank tensor in three-

dimensional Euclidean space, describing its directional asymmetry.

Let A denote a second-rank tensor in R3, which can be represented by

a 3-by-3 matrix. Assuming thatA is symmetric, A has three real eigen-

values, which are denoted by Axx, Ayy and Azz. Assume that they are

ordered such that :Axx<Ayy <Azz, then uniaxiality ∆A is represented

as 2Azz - (Axx + Ayy) and asymmetry as δA = (Ayy - Azz). In analogy

with this, the hyperfine matrix can also be used to derive uniaxiality


superconductors

and asymmetry parameters. Taking the largest diagonal element of

the A-tensor as A1 and the smallest as A3, the dipolar (anisotropic)

components can be used to derive the uniaxiality b0 = [A1 - (A2 +

A3)/2]/3 and the asymmetry c0 = (|A2| - |A3|)/2 parameters.


In this chapter, the results of the MIS and HFC calculations carried out

for the aforementioned iron compounds are presented and analysed.

The emphasis is to study the influence of the structural phase transi-

tions and the degree of doping with other elements on the hyperfine

structure parameters, MIS and HFC.

7.3.1 MIS calculations

The contact densities calculated using Eqn. 2.44 for the iron based

clusters presented above are given in Table 7.1. The isomer shifts can

be computed according to Eqn. 3.1 using the given contact densities

along with the calibration constant α(57Fe) = −0.306±0.009 a30 mm s−1

from the parametrizations in Chapter 5. Figure 7.3 plots the calculated

relative shift, ∆δx = δx - δref (in mm/s) (ref is [K4Fe(CN)6]) for both HT

and LT phases.

The isomer shifts in Table 7.1 show reasonable agreement with the

experimental shifts. It is gratifying that the calculations done using

small cluster models of the representative crystal structures could ef-

fectively reproduce the experimental results.

It is seen from the results in Table 7.1 that the overall differences

in the calculated contact densities and corresponding ∆δs within each

temperature phase is in smaller range (∆δHT ≈ 0.02 mm/s, ∆δLT ≈


Table 7.1: 57Fe electron contact densities according to Eqn. 2.44 of the ironbased superconductors.

HT LT

ρa δcalcb δexp.

b ρa δcalc.b δexp.

b

FeSe4 54.20 0.51 0.53c 54.09 0.54 0.54c

Li4FeAs4 54.22 0.49 0.44d

Ba4FeAs4 54.19 0.51 0.29e 54.02 0.56 0.42e

Sm4FeAs4O4 54.17 0.52 0.43f 54.02 0.56 0.57f

aA large constant of 15000 a0−3 has been subtracted from all the values.

bwith respect to [K4Fe(CN)6])cfrom Ref. [138]dfrom Ref. [139]efrom Ref. [24]ffor LaFeAsO, from Ref. [140]

0.02 mm/s). All the compounds with the same crystal structure have

nearly identical contact densities and isomer shifts irrespective of the

difference in the outer coordination sphere of 57Fe in different types

of superconductors, such as type 1111, 122, 111 and 011. The ba-

sic structure of these compounds is FeAs4 tetrahedra separated by

Li/Ba/SmO layers in 111, 122 and 1111 types of compounds. In the

present calculations these inter layers are represented by ECPs as ex-

plained in the earlier section, for computational simplicity. This could

be responsible for the nearly identical δs for the same crystalline phase

(HT or LT) of these compounds. Therefore, one can expect to see

a more precise distinction between different types of compounds by

computing Li/Ba/Sm with all electron basis sets.


superconductors

Figure 7.3: The ∆δ for HT (298 K) and LT (5 K) for a) FeSe4, b) Li4FeAs4, c)Ba4FeAs4 and d) Sm4FeAs4O4 .

The general trend of the isomer shift variation between the HT and

LT phase indicate that the electronic structure variations in these com-

pounds are similar (see Table 7.1 and Figure 7.3). Figure 7.3 shows a

clear jump of isomer shift with tetragonal - orthorhombic transition

in all the compounds, for FeSe4 the ∆δ is +0.05 mm/s corresponding

to a decrease of 57Fe electron contact density of ca. 0.17 a−30 with the

lowering of temperature. This feature, which is characteristic for all

the compounds studied can be suggested as a diagnostic test for the

local chemical environment of the 57Fe ions.

The superconductivity emerges in 1111 & 122 types of compounds

with the electron or hole doping of the parent compounds. Therefore,

it is interesting to study the MIS variation with the doping/chemical

substitution of these compounds. Figure 7.5 shows the MIS variation

for the HT structures with various levels of chemical substitution for

3 types of compounds, FeSe4, Ba4FeAs4 and Sm4FeAs4O4. The stud-


(a) (b) (c) (d)

Figure 7.4: The substitution of Se with S on FeSe4 clusters, (a)FeSe4 (b)FeSe3S (c) FeSe2S2 (d) FeSeS3.

ied clusters are, a) FeSe4, Se substituted partially with S, ie, FeSe3S,

FeSe2S2, FeSeS3 b) Ba4FeAs4, Ba substituted with K, Ba3KFeAs4, Ba2K2-

FeAs4, BaK3FeAs4 and c) Sm4FeAs4O4, O substituted with F, Sm4FeAs4O3F,

Sm4FeAs4O2F2, Sm4FeAs4OF3. To get a general idea of the chemical

Figure 7.5: The dependence of MIS on doping/chemical substitution

substitutions carried out in this chapter, a pictorial representation of

Se substituted with S is given in Figure 7.4. The results given in Figure

7.5 show that substitution of S for Se in FeSe4 makes a decrease in the

isomer shift, which is in the range of 0.11 mm/s. However, no such

trends are seen for Ba4FeAs4 and Sm4FeAs4O4. The reason could be

that the chemical substitution in the first coordination sphere of 57Fe


superconductors

has stronger influence on the isomer shift than that of the substitu-

tion on interlayer atoms, Ba or O. However, if one neglects the bump

in BaFeAs4 plot (Ba2K2FeAs4), the overall variation is in the decrease

of MIS with K doping which is consistent with the earlier reported

results [24].

Figure 7.5 shows that the range of variation of MIS is relatively low,

ca. 0.11 mm/s for FeSe4. In experiments, superconductivity emerges

with less than 3% of doping/chemical substitution in these compounds.

Hence this ∆δ could be smaller than 0.11 mm/s under experimental

conditions, which would indicate that the contact density variation

near 57Fe in these types of compounds is relatively low with the elec-

tron/hole doping.

7.3.2 HFC calculations

Before starting with the calculations on iron based superconductors,

the B3LYP functional and the basis sets specified earlier are tested in

the calculation of the hyperfine structure constants of complexes for

which the experimental data are available. The results along with the

references for the geometries are given in Table 7.2. In most of the

cases the calculated isotropic HFC values (Aiso) are underestimated.

The SO contributions are given for FeIISR3 and FeIIIAz. For the latter

compound, the inclusion of SO improves the agreement with experi-

ment.

The underestimation of Aiso values on compounds such as FeIIIM-

AC, FeIVMAC etc are seen by Sinnecker, Slep, Bill and Neese in Ref.

[141]. They applied a scaling factor f = 1.81 to the computed Fermi

Contact terms on the iron complexes with small values of Aiso. In

Figure 7.6, the calculated isotropic HFC is plotted against the experi-


Table 7.2: The hyperfine coupling constants (MHz) for the reference com-pounds.

Aaniso(calc.) Aaniso(exp.)

Aiso(calc.) b0 c0 Aiso (exp.) b0 c0

FeIISR3a -9.93 -3.83 0.10 -11.8 -7.3 0.00

(-8.49)b (-5.47) (0.13)

FeIIPorOAca -13.56 2.73 0.59 -21.8 1.7 0.00

FeIIIAza -15.49 -1.84 1.26 -13.5 -10.80 0.90

(-12.50) (-4.55) (1.30)

FeIIIMACa -6.77 5.01 2.55 -15.40 11.15 3.75

FeIIIPO4c -23.91 -4.21 3.68 -28.2 ... ...

FeIII(H2O)6a -19.89 -0.22 0.12 -32.60 ... ...

FeIVMACa -11.11 2.35 0.10 -20.00 2.65 2.05

aSee Ref. [141] and the references therein.bIn paranthesis, the results with SO couplingcRef. [142]

mental isotropic HFC, where for FeIISR3 and FeIIIAz, the Aiso values

represents the B3LYP values including SO couplings. For all other

complexes the B3LYP values were scaled by a factor of 1.81 (Ref. [141]).

Thus, the agreement with experimental isotropic HFC values are im-

proved.

In Table 7.2 the calculated anisotropic hyperfine coupling constants

along with experimental values are given in terms of uniaxiality b0


superconductors

Figure 7.6: Calculated 57Fe Isotropic hyperfine coupling constants vs. exper-imental hyperfine coupling constants for the reference compounds (see text

for details).

and asymmetry c0 parameters. For FeIISR3 and FeIIIAz, the inclusion

of SO corrections improves the agreement of b0 and c0 values with

experiment. The overall agreement of the calculated b0 and c0 values

with experimental values gave us confidence to use B3LYP functional

and the proposed basis sets for the calculations of the iron based su-

perconductors.

The hybrid functional with increased percentage of HF exchange,

such as BH&HLYP are tested on selected compounds in order to anal-

yse their performance. The results showed an increase of the isotropic

HFC values of ca.4-5 MHz in magnitude and hence it can improve the

agreement with experimental values. However, the SO contribution

within the BH&HLYP functional for one of the reference compound

produces a higher positive value, making the overall isotropic HFC

value similar in magnitude to the B3LYP result. Therefore, the calcu-

lations here are done within B3LYP functional only.


The HFCs with and without SO corrections are given in Table 7.3

for the considered iron based superconductors. For all the compounds,

there is a slight decrease of the Aiso values with the tetragonal-orthorh-

ombic phase transition. The trend is similar for the Aiso values includ-

ing SO corrections.

The isotropic HFC values (Aiso) without including SO corrections

differ by ca.1 MHz for the HT and LT phases (see Table 7.3). How-

ever, the SO corrections bring in a clear distinction between the dif-

ferent types of compounds. Based on the results, the iron-based su-

perconductors can be divided into 3 groups, a) FeSe4 and Li4FeAs4

b) Ba4FeAs4 and c) Sm4O4FeAs4. The first group which has relatively

short interlayer separation (see Figure 7.1(a,b)), has lower Aiso values,

whereas for b and c groups the Aiso values increase. With the increase

in the inter layer separation, from 011 to 1111 types, the spin density

near the 57Fe increases, which is evident from the calculations includ-

ing SO corrections. The SO contributions account for up to 30 % of the

hyperfine coupling constant, therefore it is crucial for the understand-

ing of the magnetic hyperfine coupling constants of these compounds.

The anisotropic contributions of the hyperfine coupling constants

represented by the uniaxiality b0 and asymmetry c0 are given in Ta-

ble 7.3. The results show that there is an increase of the uniaxiality in

all the compounds with the tetragonal-orthorhombic phase transition.

The asymmetry c0 increases from 011 to 1111 types (see Table7.3), in-

dicating a decrease of the tetrahedral symmetry of FeX4 (X = Se/As)

in these compounds from 011 to 1111 types.

An apparent small magnetic moment per iron atom (µ ≈ 0.4µB)

was reported for pnictides of the 1111 type, which is much less than

in 122 compounds (≈ 0.9 µB) and they are much smaller than the DFT


superconductors

Table 7.3: Hyperfine coupling constants (MHz) for the iron compoundsstudied both including and excluding the SO effects at B3LYP level of the-

ory.

Aaniso Aaniso(SO)

Aiso b0 c0 Aiso (SO) b0 c0

FeSe4-HT -10.93 -4.14 0.01 -7.77 -5.86 0.01

FeSe4-LT -10.55 -4.18 0.17 -7.50 -6.04 0.01

Li4FeAs4-HT -11.90 -3.32 0.44 -8.61 -4.71 0.525

Ba4FeAs4-HT -11.51 5.08 0.95 -9.31 5.78 2.625

Ba4FeAs4-LT -10.96 5.13 1.27 -9.03 6.09 2.785

Sm4O4FeAs4-HT -11.86 3.17 2.09 -10.40 4.33 2.33

Sm4O4FeAs4-LT -11.33 3.25 2.98 -10.02 4.84 2.61

predicted value (≈ 2 µB) [144, 145]. The difference in the magnetic

moment between 1111 type and 122 type could be that the interlayer

coupling of FeAs layers is stronger in 122 compounds than 1111 com-

pounds because the interlayer distance in 122 compounds is signifi-

cantly shorter than 1111-compounds (see Figure 7.1). There are sev-

eral factors contributing to the reduction of the observed magnetic

moment, among which can be mentioned : i) the effect of itinerant

electrons, the DFT calculations are done on a small magnetic unit cell

and the interactions with the itinerant electrons are not considered,

therefore the moment obtained by DFT is the bare moment of each Fe

ion, and ii) the effect of SO coupling, the SO coupling induces singlet-

triplet mixing which can reduce the observed magnetic moment per


iron atom. In the present calculations, it is seen that the influence of

SO corrections on Aiso values are relatively strong (up to ca 30 %) and

there is a reduction of the absolute values indicating a reduction in the

spin density (see Table 7.3). Therefore, it can be inferred that with the

inclusion of SO corrections, there is singlet-triplet mixing and corre-

sponding decrease in spin density.

The effect of chemical substitution or electron doping on the calcu-

lated isotropic hyperfine coupling constants are evaluated for a num-

ber of representative clusters of a) FeSe4 and b) Li4FeAs4 and c) Sm4O4-

FeAs4. For FeSe4, partial substitution of Se with isovalent S atoms, ie,

FeSe3S, FeSe2S2; Li4FeAs4 where a partial substitution of Li with K,

Li3KFeAs4 and Li2K2FeAs4; Sm4O4FeAs4 where a partial substitution

of O with F, Sm4O3FFeAs4 and Sm4O2F2FeAs4 are carried out. The

results are plotted in Figure 7.7. Similar to the MIS calculations, sub-

stitution of the interlayer atoms, ie, O with F in Sm4O4FeAs4 and Li

with K in Li4FeAs4, does not show a clear trend in the Aiso variations,

whereas the substitution in the direct co-ordination sphere of 57Fe, ie,

Se with S makes a slight increment in the magnitude of Aiso values.

All these three classes (a-c) show isotropic HFC values (Aiso) within

0.25 MHz variation with the chemical substitutions, suggesting an in-

dependence of Aiso values on the degree of chemical substitution.

However, the inclusion of SO corrections for Li3KFeAs4 gives an

Aiso value of -9.38 MHz, whereas the Aiso for Li4FeAs4 is -8.61 MHz,

which makes a noticeable increase in the magnitude of the hyperfine

coupling constant indicating an increase of spin density near 57Fe nu-

cleus with the K- doping. Therefore, it is clear that the distortion in the

local geometry and changes in the spin density near 57Fe with the elec-

tron/hole doping in these compounds can only be accurately studied

by the calculations including the SO corrections.


superconductors

Figure 7.7: The dependence of Aiso on doping/chemical substitution

7.4 Conclusion

The knowledge of the local electronic structure in iron-based super-

conductors should help to understand the interplay between the struc-

ture and the emergence of superconductivity in these compounds. In

the present chapter, the iron based high temperature superconductors,

FeSe, LiFeAs, BaFe2As2 and SmOFeAs are investigated in order to un-

derstand the variation of the isomer shifts and the hyperfine coupling

constants with tetragonal-orthorhombic phase transition.

The isomer shift calculations within the linear response approach

have been done using double hybrid density functional method (B2-

PLYP). The relative 57Fe isomer shifts calculated with respect to [K4Fe-

(CN)6] for tetragonal/orthorhombic crystal structure are within a vari-

ation of less than 10% irrespective of the different chemical environ-

ments. The calculations have shown that the electron contact density

near the 57Fe nucleus reduces and correspondingly the isomer shift

increases with the tetragonal - orthorhombic phase transition. This

7.4. Conclusion 131

variation can be taken as a criterion for monitoring the local chemical

environment around the resonating nuclei, in different temperature

phases and during phase transitions.

The hyperfine coupling constant calculations done with the B3LYP

functional show that the tetragonal-orthorhombic transition produces

only a slight variation in the local spin moment near the nucleus,

and consequently in the isotropic hyperfine coupling constants, Aiso.

However, including the SO corrections in the calculations reduces the

spin density near the 57Fe nucleus, and the Aiso values. The SO cor-

rections scale up to 30 % of the calculated Aiso values, making a dis-

tinction between different types of the superconductors. It has been

observed that with the increase in the interlayer separation, ie, from

011 to 1111 types, there is an increase of Aiso values. The uniaxial-

ity of the A-tensor of these compounds increases with the tetragonal-

orthorhombic phase transitions.

The dependence of MIS and HFC on doping has shown that the

influence is stronger with the chemical substitution in the immedi-

ate coordination sphere of 57Fe and it is less noticeable with the dop-

ing/chemical substitution in the inter layers such as Li/Ba/SmO in

111, 122, 1111 types of compounds.

The present chapter investigates on the local chemical environ-

ment near the 57Fe nucleus in the iron based superconductors within

the scope of the Mossbauer spectroscopy parameters, the isomer shift

and hyperfine coupling constant. The results indicate that these (MIS

and HFC) are good tools to get an insight into the variation of the elec-

tronic structure and magnetic spin densities under phase transition

and upon doping/chemical substitutions in the coordination sphere

of 57Fe. Therefore, these tools can effectively be used for monitor-

ing the structure of these compounds and in a wider perspective con-


superconductors

tribute to a better understanding of the mechanism behind supercon-

ductivity in these compounds.

Chapter 8

Conclusions and Outlook

Synopsis

In the present chapter, the general conclusions of this work are drawn and

some suggestions for the future work are presented.

8.1 Conclusions

The purpose of this thesis project was to use the recently devel-

oped linear response formalism, which incorporates the impor-

tant effects such as relativity and electron correlation, for the the the-

oretical modeling of Mossbauer isomer shift. The theoretical descrip-

tion of MIS requires a reliable value of the calibration constant α,

which is usually obtained from the linear regression of theoretical con-

tact densities against the experimental isomer shifts. It was our hope

that the use of the highly accurate theoretical methods in combination

with the linear response approach would provide a universally ap-

plicable value of the calibration constant α for 57Fe and 119Sn nuclei,

which could be used in the future MIS calculations on various 57Fe

and 119Sn based systems without the necessity to parametrize the cal-

ibration constant for the given method. The major conclusions drawn

from this dissertation are,

134 8. Conclusions and Outlook

Chapter 3

• The performance of various density functionals within the linear re-

sponse formalism for the MIS calculations were analyzed, the results

pointed out that the correlation of calculated MIS with the experimen-

tal values improves with the increased fraction of the HF exchange in

the density functional.

• The use of standard contracted basis sets yields reasonable results at

the HF and density functional levels of theory, however with the use

of the highly correlated ab initio methods the small basis sets lead to

substantial errors, suggesting the need to use the extended basis sets

in connection with high-level ab initio methods.

• The MIS calculations carried out using a literature value of the cali-

bration constant α does not produce satisfactory correlation with the

experimental isomer shifts even when the hybrid functionals are used

in combination with large basis sets, therefore, it is suggested to use

a constant α obtained from the calibration of the contact densities cal-

culated with the most accurate theoretical methods against the exper-

imental isomer shifts.

Chapter 4

• The importance of proper description of relativity and electron cor-

relation for the accurate determination of contact density near the nu-

cleus is demonstrated (see Figure 4.2).

• The use of highly correlated ab initio methods gives a better de-

scription of the electron contact density near the 119Sn nucleus and

improves the calibration constant α.

• The calibration constant αMP2(119Sn) = (0.091 ± 0.002)a0

−3 mm/s ob-

tained with the use of relativistically corrected MP2 calculations is in

excellent agreement with the previously obtained values, and can be

proposed as a reliable, universally applicable constant for future MIS

8.1. Conclusions 135

calculations thus eliminating the necessity to develop method depen-

dent parametrizations.

• The MIS calculations on CaSnO3 perovskite for the pressure range

0-36 GPa yield isomer shifts in a reasonable agreement with the ex-

perimental results. This confirms that the linear response formalism

is capable of describing small variations of the isomer shift resulting

e.g. from the pressure variations or from phase transitions.

Chapter 5

• The improved correlation with the experiment obtained by the use

of the highly correlated methods (which is seen in Chapter 4), moti-

vates us to test the double hybrid density functional developed by S.

Grimme [55] in the calibration of α(57Fe).

• The calibration constant, αB2−PLYP = −0.306 ± 0.009 a30 mm s−1 ob-

tained gives the best statistical correlation with the experimental data,

and this value is in excellent agreement with the experimental esti-

mate of α = −0.31± 0.04 a30 mm s−1. This suggests to use this constant

for the future 57Fe MIS calculations.

Chapter 6

• The presence of different iron sites observed experimentally in the57Fe Mossbauer spectra of Prussian blue analogue RbMn[Fe(CN)6]·H2O

is analysed using the 57Fe parametrization developed in Chapter 5.

• The isomer shift differences ∆δHT−LT of ∼ 0.1 mm/s between the HT

and LT phase of the RbMn[Fe(CN)6]·H2O compound originate from

the change of oxidation state of iron from FeIII to FeII.

• The ∆δ differences of 0.01-0.03 mm/s of the two different iron sites

within both HT and LT phases originate from the different distribu-

tion patterns of Rb1 ions around the iron sites.

Chapter 7

• The recently discovered iron based superconductors are studied in


order to analyse the variation of the hyperfine structure parameters;

the isomer shift and hyperfine coupling constant, under phase tran-

sition and upon chemical substitution of these compounds. The MIS

calculations are based on the parametrization developed in Chapter 5.

• The MIS study shows that with the tetragonal-orthorhombic phase

transition in these compounds, the contact density decreases (ca. 0.17

a−30 ) and the isomer shift increases (ca. +0.05 mm/s), which is charac-

teristic for all the iron based superconductors studied.

• The hyperfine coupling constants calculated with the inclusion of

the spin-orbit corrections give distinct variations among different classes

of the iron based superconductors. The spin density near the 57Fe nu-

cleus increases with the increase in the inter layer separation in the

geometry, ie when going from 011 to 1111 types of compounds.

• The dependence of MIS and HFC on doping/chemical substitutions

shows that its influence is stronger when the chemical substitution

occurs in the immediate coordination sphere of 57Fe and it is less no-

ticeable with the doping/chemical substitution in the inter layers such

as Li/Ba/SmO in 111, 122, 1111 types of compounds.

• The Mossbauer parameters, MIS and HFC, are evaluated to be good

tools to study the variation of the electronic structure and magnetic

spin densities under phase transition and upon doping/chemical sub-

stitutions in the coordination sphere of 57Fe for the studied iron based

superconductors, and can be suggested for monitoring the structural

changes in these compounds.

Based on the analysis of the performance of different quantum

chemical computational schemes in the calibration of the MIS, reli-

able and universally applicable values of the calibration constant α for57Fe and 119Sn are obtained. The use of these values should eliminate

the necessity for a tedious parametrization procedure when studying

8.2. Outlook 137

the parameters of Mossbauer spectra. The application of the obtained

calibration constants in combination with accurate quantum chemi-

cal methods helped us to obtain interesting results in the study of the

Prussian blue analogues and of the iron-containing superconductors.

8.2 Outlook

As possible directions for the future work, the following points can be

suggested,

Parametrizations of other elements.

The reliable values of the calibration constant α for 57Fe and 119Sn ob-

tained in this project, suggest to use the linear response formalism for

the determination of accurate values of α for other Mossbauer active

nuclei in the periodic table, to name a few, Ta, Eu, Au, Hg, I, U etc.

Generally, the theory level for the MIS calculations are chosen by

analysing the linear regression of the calculated isomer shifts within

a particular theoretical method against the experimental shifts of the

chosen compounds. The initial task in this direction is to yield a

calibration constant α from the empirical fit of the contact densities

against the exeperimental isomer shifts. Therefore, if an accurate,

method independent calibration constant α is available then one can

use it directly and avoid the parametrization of the calibration con-

stant for every particular method of calculation, which saves consid-

erable time.

Development of basis sets for MIS calculations.

All the calculations in this dissertation are carried out using the stan-

dard large uncontracted basis sets. However, the calculation using

these basis sets are complex and time consuming. Therefore, it is es-


sential to investigate the possibility to construct more compact, yet

accurate basis sets for the MIS calculations. That such a possibility ex-

ists is suggested by the results of Chapter 3, where good results were

obtained with the standard contracted basis sets of small and medium

size. The development of the compact basis sets for MIS calculations

will enable us to save the computational work and to study bigger

systems.

Applications in the fields of Materials science and Biological Sci-

ences.

• Materials Science

There are a number of aspects which can be studied using Mossbauer

technique, in materials containing Mossbauer active nuclei. They in-

clude, the valence state determination, structural and bonding prop-

erties, solid state reactions, spin crossover, magnetic crossover etc.

Mossbauer spectroscopy is extensively used to study the valence

fluctuations and temperature dependent electron transfer between dif-

ferent metals in mixed valence metallic compounds. The capability of

MIS to predict the valence state in such systems is clear from the anal-

ysis of the Prussian blue analogue, RbMn[Fe(CN)6]·H2O, described in

Chapter 6. The Mossbauer parameters can effectively be used for the

study of the electronic structure of newly discovered materials, for ex-

ample the iron based superconductors, which is explained in Chapter

7 of this dissertation.

• Biochemistry and Biocatalysis

There are numerous proteins which contain iron as their active site,

the functional properties of these proteins are determined mostly by

the iron electronic structure. Therefore, the small changes in the iron

electronic structure under various conditions can be detected by Mos-

sbauer spectroscopy. For instance the analysis of oxyhemoglobins

8.2. Outlook 139

from variuos types of leukemia patients show a slight increase of the

Mossbauer parameters, the isomer shift and quadrupole splitting, in

comparison with the normal adults [146]. Moreover, several diseases

are caused by the synthesis of anomalous biomolecules or from the

disturbance in the biosynthesis of any other proteins. In these cases,

the Mossbauer spectroscopy is helpful to study the electronic struc-

tural variations of these iron containing proteins.

Iron-oxo species are the key reactive intermediates in the cataly-

sis of oxygen-activating enzymes and synthetic catalysts, such as cy-

tochrome P450 and methane monooxygenase (MMO). High valent

iron-oxo intermediates are used in biological systems to carry out chal-

lenging oxidations in many scenarios. The analysis of the Mossbauer

properties of these intermediate states in the reaction cycle, brings an

understanding of the oxidation activation in these enzymes.

There are several metabolic reactions/pathways, which can be stud-

ied using Mosbauer spectroscopy. Certain disturbances in the metabolic

pathways can cause several anomalies, for example the overload of

iron, resulting in diseases such as β-thalassemia, hemochromatosis

etc. Therefore, the electronic and magnetic structure of Mossbauer el-

ements in the metabolic reactions/pathways can give information on

the molecular nature of the diseases and pathological processes occur-

ring in the living organisms.

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Summary

The Mossbauer spectroscopy parameters such as the isomer shift, quadru-

pole splitting and magnetic hyperfine splitting carry substantial information

on the electronic structure, spin- and oxidation states of the resonating atom.

This dissertation is focused mainly on the isomer shifts, which is a measure

of energy difference between the energies of γ-transitions occurring in the

sample (absorber) nucleus compared to the reference (source) nucleus. Be-

cause the electronic environments in which the sample and the reference nu-

clei immersed are different, the isomer shift probes this difference. However

the interpretation of the relationship between the isomer shift and the elec-

tronic structure is not straight forward and the first principle models are es-

sential. The existing traditional perturbational treatment of the isomer shifts

are not very accurate and lack some essential characteristics.

This dissertation details a new approach developed recently [J.Chem.Phys.

127, 084101 (2007)], within which the isomer shift is expressed as the deriva-

tive of electronic energy with respect to the nuclear radius, it has the clear

advantage that relativistic and electron correlation effects can be directly in-

cluded in the calculation. With the use of the linear response formalism,

the performance of various quantum chemical methods were analysed and

reliable parametrization of 57Fe and 119Sn nuclei were obtained, which are

detailed in this dissertation. The application of the calculated calibration

parameters for the investigation on various iron based systems such as the

Prussian blue analogues and the recently discovered high temperature su-

perconductors are described in this dissertation.

Samenvatting

De Mossbauerspectroscopische parameters, zoals de isomerieverschuiving,

quadrupoolsplitsing en magnetische hyperfijnsplitsing dragen substantile

informatie over de elektronische structuur, spin- en oxidatietoestanden van

het resonerende atoom. Dit proefschrift is vooral gericht op de isomeriever-

schuivingen, die een maat zijn van de verschillen tussen de energieen van

γ-overgangen binnen de kern in het monster (absorber) in vergelijking met

de referentiekern (bron). Omdat de elektronische omgevingen waarin de

monster- en referentiekern zich bevinden anders zijn, is de isomerieverschuiv-

ing een maat voor dit verschil. Echter, de interpretatie van de relatie tussen

de isomerieverschuiving en de elektronische structuur is niet eenvoudig en

modellen gebaseerd op eerste beginselen zijn essentieel. De bestaande tradi-

tionele modellen gebaseerd op storingsrekening zijn niet erg nauwkeurig en

missen een aantal essentile kenmerken.

Dit proefschrift beschrijft in detail een onlangs ontwikkelde nieuwe aan-

pak [J.Chem.Phys. 127, 084101 (2007)], waarbinnen de isomerieverschuiving

wordt uitgedrukt als de afgeleide van de elektronische energie naar de straal

van de kern. Het heeft het duidelijke voordeel dat relativiteit en correlatie-

effecten direct kunnen worden opgenomen in de berekening. Met het ge-

bruik van het lineaire-responsformalisme werden de prestaties van verschil-

lende kwantumchemische methoden geanalyseerd en betrouwbare param-

eterisaties van 57Fe- en 119Sn-kernen verkregen, die zijn beschreven in dit

proefschrift. De toepassing van de berekende kalibratie parameters op het

onderzoek van verschillende op ijzer gebaseerde systemen, zoals Pruisisch-

blauwanalogen en recentelijk ontdekte hoge-temperatuur supergeleiders, is

uitgevoerd en beschreven in dit proefschrift.

List of Publications

1. Calibration of 57Fe isomer shift from ab initio calculation: Can theory and

experiment reach an agreement?, R. Kurian and M. Filatov, Phys. Chem.

Chem. Phys., 12, 2758 (2010). (Chapter 5)

2. Theoretical investigation on the presence of different iron sites in RbMn[Fe-

(CN)6].H2O, R. Kurian and M. Filatov, J. Phys. Conf. Series, 217, 012012

(2010). (Chapter 6)

3. Calibration of 119Sn isomer shift using ab initio wave function methods, R.

Kurian and M. Filatov, J. Chem. Phys., 130, 124121 (2009). (Chapter 4)

4. DFT Approach to the calculation of Mossbauer Isomer Shifts, R. Kurian and

M. Filatov, J. Chem. Theory Comput., 4, 278 (2008). (Chapter 3)

5. Dependence of 29Si NMR chemical shielding properties of precursor silicate

species, Q0 on its local structure at the pre-nucleation stages of zeolite synthesis-

A DFT based computational correlation, K. Selvaraj and R. Kurian, Micro-

porous and Mesoporous Materials, 122, 105, (2009). (not within the scope

of this thesis)

Acknowledgements

It would not have been possible to write this doctoral thesis without the help

and support of the kind people around me, to only some of whom it is pos-

sible to give particular mention here.

First and foremost, I thank my supervisor Prof. Michael Filatov for his

patience, motivation, enthusiasm and immense knowledge. Throughout my

doctoral years he provided good teaching and sound advice which enabled

me to complete this work. I can not imagine having a better advisor and

mentor for my Ph.D study than him and I am indebted to him more than he

knows. I would like to thank Michael’s wife Inna for her care and kindness

especially during my first days in Netherlands.

I am thankful to Prof. Ria Broer for her fruitful suggestions and invalu-

able care throughout my PhD period. I also would like to thank Prof. Wim

Nieuwpoort for the comments and suggestions during various group semi-

nars and discussions.

For this dissertation I would like to express my deep and sincere grati-

tude to the reading committee members: Prof. Franck Neese, Prof. Lucas

Visscher, and Prof. Yann Garcia for their time, interest, and helpful com-

ments. I also would like to thank Prof. Beatriz Noheda, Prof. Rob de Groot,

Dr. Stefaan Cottenier and Dr. Vladimir Malkin for being in my defense com-

mittee.

I express my sincere gratitude to Prof. Martin Kaupp for allowing me

to work in his group more than a month which shaped into a nice piece of

work in this thesis. I enjoyed the stay in Wurzburg and the fruitful discus-

sions with Martin and his group members, especially Florian, Manuel and

Johannes.

I am indebted to the past and present members of Theoretical Chemistry

Group. Andranik, Andrii, Dani, Daniel, Gerrit-Jan, Hilde, Johan, Muizz,

Remco, Rob, Rodrigo for their friendship which created a pleasant working

atmosphere. My special thanks to Rob for a quick translation of the Sum-

mary of this thesis to Dutch and also for being one of my paranymphs. An-

dranik, it was a huge relief to share the progress of thesis, as both of us were

in the same boat. I thank Johan for helping with all the technical problems.

Some of the past group members that I have had the pleasure to work with

or alongside of are Olena and Aymeric.

I would like to thank Dr. K. Selvaraj for introducing me to the world of

scientific research. During the six months stay for my master’s project in his

group at the NCL, Pune, I got inspired to get into the scientific field.

My time in Groningen was made enjoyable in large part due to the many

friends who became a part of my life. I feel to express thanks to Achus,

Aziz, Bahju, Durba, Mariam, Sivji, Nibu, Sherin, Shamz for their invaluable

friendship. My special thanks to Durba for being one of my paranymphs.

I cannot conclude without thanking my family, on whose love and en-

couragement I have relied throughout my life. I would like to thank my

father K. M. Kurian, mother Marykutty, brother Rajesh, sister in law Tincy

and dearest little ones, Serah and Erick. Your support, prayers and care have

taken me to new stages of life, for which my mere expression of thanks like-

wise does not suffice. I would like to extend my thanks to my in laws, M.

V. John and Annamma, my brother in law and family, Gijo, Mickey and Ben,

for all the love and care they have shown to me.

I am extremely lucky to have Charly in my life, more than just a husband

you are my best critic and best friend. Your professionalism and dedication

to work motivated me lot, your passion for the open source technologies

keeps me updated in the same, your corrections helped me to improve my-

self in all the ways, even if I did not show this to you :).

Once again I thank everybody who was important to the successful com-

pletion of this thesis.

Reshmi Kurian

University of Groningen First principles theoretical ... · Chapter 1 Introduction and Objective...

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