164 BUNSEN-MAGAZIN · 15. JAHRGANG · 4/2013 UNTERRICHT METHOD SUMMARY Acronyms - Variable Temperature-Variable Field-Magnetic Circular Dichr- oism (VTVH-MCD) - Optical Rotatory Dichroism (ORD) - Circular Dichroism (CD) - Zero Field Splitting (ZFS) - Electron Paramagnetic Resonance (EPR) - Left/right circularly polarized (lcp/rcp) - X-ray Magnetic Circular-/X-ray Magnetic Linear Dichroism (XMCD/XMLD) Benefits (Information Available) - Ground and excited state spin and orbital degeneracy - Spectral band polarization - Ground and excited state magnetic dipole moment, g-fac- tors, oxidation state - Coordination number - Axial and rhombic Zero Field Splitting - Allows assignment of electronic transitions - Improved resolution compared to absorption measurements - Good for metalloenzyms/-proteins with small amount of par- amagnetic centers Limitations - Metal coordination may be only indirectly determined - Samples must be optically transparent - Not useful for non chromophoric metal centers 1. INTRODUCTION The aim of this contribution is to bring the Magnetic Circular Di- chroism (MCD) to non-specialists or beginners. MCD is the dif- ference of absorption of left and right circularly polarized light caused by the external magnetic field applied parallel to the di- rection of the light propagation. This is a research method, used for studies of the role of transition metal centers in chemical compounds. These metal centers determine electronic, mag- netic and structural properties important for several applica- tions and for understanding catalytic or biochemical processes. MCD may be used as a stand-alone method but usually comple- mentary methods are used together with MCD to probe the elec- tronic and magnetic properties. Broad information is provided by the optical absorption spectroscopy monitoring the d-d transi- tions in the visible range mainly and charge-transfer processes in the Ultraviolet-Visible (UV-VIS) range. For characterizing the magnetic properties, which reflect the ground state electronic structure, magnetic susceptibility and magnetization measure- ments are indispensable. However, to determine the exact val- ues of electronic parameters for the ground state in complexes the frequently used method is the Electron Paramagnetic Reso- nance (EPR) spectroscopy. The High Field Electron Paramagnetic Resonance, in turn, allows in the easy way to monitor a large Zero Field Splitting. The electronic parameters experimentally obtained may be used for verification of theoretical models build on the base of Molecular Orbital (MO) or extended Density Func- tional Theory (DFT) methods. Wolfgang Haase 1 , Serghei M. Ostrovsky 1,2 , Zbigniew Tomkowicz 1,3 CHARACTERIZATION OF ELECTRONIC AND MAGNETIC PROPERTIES OF TRANSITION METAL COMPLEXES BY MAGNETIC CIRCULAR DICHROISM SPECTROSCOPY 1 Eduard-Zintl-Institute of Inorganic and Physical Chemistry, Technical University Darmstadt, Petersenstr. 20, 64287 Darmstadt, Germany 2 Institute of Applied Physics, Academy of Sciences of Moldova, Academy Str. 5, MD-2028 Chisinau, Moldova 3 Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland Prof. Dr. Wolfgang Haase E-Mail: [email protected]Dr. Serghei Ostrovsky E-Mail: [email protected]Dr. hab. Doz. Zbigniew Tomkowicz E-Mail: [email protected]
oism (VTVH-MCD)- Optical Rotatory Dichroism (ORD)- Circular Dichroism (CD)- Zero Field Splitting (ZFS)- Electron Paramagnetic Resonance (EPR)- Left/right circularly polarized (lcp/rcp)- X-ray Magnetic Circular-/X-ray Magnetic Linear Dichroism
(XMCD/XMLD)
Benefi ts (Information Available)- Ground and excited state spin and orbital degeneracy- Spectral band polarization- Ground and excited state magnetic dipole moment, g-fac-
tors, oxidation state - Coordination number- Axial and rhombic Zero Field Splitting - Allows assignment of electronic transitions- Improved resolution compared to absorption measurements- Good for metalloenzyms/-proteins with small amount of par-
amagnetic centers
Limitations - Metal coordination may be only indirectly determined- Samples must be optically transparent- Not useful for non chromophoric metal centers
1. INTRODUCTION
The aim of this contribution is to bring the Magnetic Circular Di-chroism (MCD) to non-specialists or beginners. MCD is the dif-ference of absorption of left and right circularly polarized light caused by the external magnetic fi eld applied parallel to the di-rection of the light propagation. This is a research method, used for studies of the role of transition metal centers in chemical compounds. These metal centers determine electronic, mag-netic and structural properties important for several applica-tions and for understanding catalytic or biochemical processes.
MCD may be used as a stand-alone method but usually comple-mentary methods are used together with MCD to probe the elec-tronic and magnetic properties. Broad information is provided by the optical absorption spectroscopy monitoring the d-d transi-tions in the visible range mainly and charge-transfer processes in the Ultraviolet-Visible (UV-VIS) range. For characterizing the magnetic properties, which refl ect the ground state electronic structure, magnetic susceptibility and magnetization measure-ments are indispensable. However, to determine the exact val-ues of electronic parameters for the ground state in complexes the frequently used method is the Electron Paramagnetic Reso-nance (EPR) spectroscopy. The High Field Electron Paramagnetic Resonance, in turn, allows in the easy way to monitor a large Zero Field Splitting. The electronic parameters experimentally obtained may be used for verifi cation of theoretical models build on the base of Molecular Orbital (MO) or extended Density Func-tional Theory (DFT) methods.
Wolfgang Haase1, Serghei M. Ostrovsky1,2, Zbigniew Tomkowicz1,3
CHARACTERIZATION OF ELECTRONIC AND MAGNETIC PROPERTIES OF TRANSITION
METAL COMPLEXES BY MAGNETIC CIRCULAR DICHROISM SPECTROSCOPY
1 Eduard-Zintl-Institute of Inorganic and Physical Chemistry, Technical University Darmstadt, Petersenstr. 20, 64287 Darmstadt, Germany
2 Institute of Applied Physics, Academy of Sciences of Moldova, Academy Str. 5, MD-2028 Chisinau, Moldova
3 Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland
MCD is similar to the Faraday effect. The Faraday effect de-scribes the rotation of the plane of light polarization resulting from optical activity induced by the magnetic fi eld oriented parallel to the propagation of the light beam, far from the absorption bands. The Faraday effect is observed for every substance. The MCD itself appears in absorbing media at the absorption band as a result of the Zeeman effect, i.e. energy levels splitting in the magnetic fi eld and/or the magnetic fi eld mixing of electronic states. This is in contrast to the natural Circular Dichroism (CD) observed for a substance with natural optical activity. MCD is not new and the fundamentals of the theory go back to the sixties of the previous century, but over the last decade this method has undergone a renaissance. The modern MCD apparatus is equipped with superconducting magnet, so that measurements in high fi elds up to 12 T and low temperatures down to 1.3 K and lower are possible. The MCD method with such possibilities is called Variable Temper-ature-Variable Field (VTVH-)-MCD.
Within this contribution we present the basic description of the MCD method. We show also how parameters of the electronic ground and excited states, such as spin-orbit interaction, Zero Field Splitting or exchange coupling can be obtained. We will restrict ourselves to the electronic and magnetic properties of paramagnetic metal complexes (open shell systems). The use-fulness of the VTVH-MCD method in comparison with UV-VIS absorption and magnetic measurements is demonstrated.
The circularly polarized synchrotron light allows exercising simi-lar experiments using the X-ray beam. Our presentation ends with a concise description of the X-ray MCD method.
2. ELECTRONIC TRANSITIONS WITHOUT AND WITH APPLIED MAGNETIC FIELD
2.1. POLARIZED LIGHT
Light represents a transverse electromagnetic wave with elec-tric and magnetic fi elds oscillating perpendicular to the direc-tion of the wave propagation and perpendicular to each other. Since the electric and magnetic fi eld vectors are in phase, only the electric fi eld vector is dealt with. If the electric fi eld of the electromagnetic wave oscillates only in one plane (along the
light propagation) the light is considered to be linear (or plane) polarized. This plane is assigned as a plane of polarization. Let us regard a superposition of two electromagnetic waves, one polarized in the xz plane and another in the yz plane, z being the direction of the light propagation. Depending on the phase differences of two regarded waves, different situations occur. When the phase difference is one-quarter of a wavelength, the resultant electromagnetic wave is circularly polarized (Fig. 1).
The electric fi eld vector travels in space along the thread of the right-hand screw for the right circularly polarized (rcp) electro-magnetic wave and the left-hand screw for the left circularly polarized (lcp) one.
The electric fi eld vector of a circularly polarized light travelling in the positive z direction can be written as
� � � �� �czntiEi /2exp2
1Re 0 �� ��� jiE , (1)
where the subscripts + and – refer to right and left circularly polarized light, respectively, i and j are unit vectors along the x and y axes, ν is the frequency, t is the time, n± is the refractive index of the medium and c is the velocity of light in vacuum. Re means real part. Eq.(1) can be rewritten as
� �� � � �� �� �czntczntE /2sin/2cos2
10 ��� ��� jiE � . (2)
To illustrate the behavior of the electric fi eld vector of a circularly polarized light, one can write Eq.(2) in a polar coordinate system in the xy plane with x axis being the polar one. The radius and polar angle are 2/0E�� and � �cznt /2 ��� ' � , respec-tively, with upper sign in ϕ corresponding to the right circularly polarized light and lower one to the left circularly polarized light. It is seen that the electric fi eld vector rotates in the xy plane clockwise and anti-clockwise as viewed in –z direction (from the receiver to the light source) for the right and left circularly polar-ized light, respectively (Fig. 2).
If both rcp and lcp light travels through the isotropic medium (n+ = n-) the resulting wave is plane polarized in the x direction (Fig. 2, left). Thus, two plane polarized waves with one-quarter of a wavelength phase difference can be combined to give circularly
Fig. 1: Right circularly polarized electromagnetic wave as a superposition of the xz and yz plane polarized waves shifted by one-quarter of a wavelength.
Fig. 2: Left: Behavior of the electric fi eld vector for the right (red) and left (blue) circularly polarized electromagnetic waves. Resulting plane polarized wave is green. Right: Rotation of the plane of polarization for an anisotropic medium (n+ ≠ n-). Viewed in –z direction (from the receiver to the light source).
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polarized wave and vice versa two circularly polarized waves can give plane polarized wave. If the medium is anisotropic (n+ ¹ n-) the resulting plane polarized wave will be rotated relative to the initial plane of polarization (Fig. 2, right). The angle of rotation is propor-tional to the travelling distance of the wave and the difference of refraction indexes. If the medium absorbs light and this absorp-tion depends on the light polarization then the amplitudes of the polarized components are no longer equal to each other and the elliptically polarized light goes out from the medium (Fig. 3).
Essentially the CD effect is connected to chiral centers (pack-ing in the crystal structure may be also essential, as in quartz) being absent for non chiral systems. Insofar the CD measure-ments provide valuable information for all chiral molecules including proteins/enzyms or biocatalytical processes, where chirality is involved or changed. CD measurements on metal containing chiral systems help to evaluate the absolute con-fi guration of the species and the secondary structure of chiral metalloproteins. For those the fraction of a molecule, being in the alpha-helix conformation or in some other conformations, can be clearly evaluated [1].
CD spectra provide much more detailed information with re-spect to conformational changes of the entire molecule than absorption spectra do; this is because of the individual sign of lcp and rcp components of the spectra.
2.3. MCD SPECTROSCOPY
2.3.1. FARADAY A, B AND C TERMS
In this section the main equations and formulas of the MCD theory (formalism) are presented. Readers interesting in the details of derivations are referred to [2-7]. At the assumption that Zeeman splitting is small compared to kBT (with kB and T being the Boltzmann constant and temperature, respectively) and to the bandwidth of the absorption line, the MCD signal for the a ® b absorption band can be written as [7]
��
���
����
����
����
��
���
��
��
)()( 0
01 EfTk
CBEEfAB
EA
BzB�� , (3)
where DA is the difference in absorbance of lcp and rcp light, E is the energy of the incident radiation, g is some constant that depends on the studied sample and f(E) is a normalized band shape function. mB and Bz are Bohr magneton and the induction of the external magnetic fi eld applied along z-direction, respec-tively. In the derivation of eq. (3) different approximations have been used, namely, (i) the Born-Oppenheimer approximation that the wave function can be written as a product of electronic and nuclear (vibronic) parts, (ii) Franck-Condon approximation that an electronic transition takes place without changes in the nuclear positions and (iii) the rigid-shift approximation that the applied magnetic fi eld changes the position of the absorption line but not the shape of this line.
As can be seen from eq. (3), the MCD signal consists of three contributions (so called A, B and C terms). B- and C-terms both show an absorption band shape, whereas the A-term corre-sponds to a signal with a derivative band shape. The A-term appears in the MCD spectrum due to the change of the position of the absorption lines in the presence of the magnetic fi eld. The corresponding A1 parameter is
*� �,
1 221 ����
��
aSLabSLbd
A zzzza
���� �
� �22���� bmabma �� � .
(4)
Fig. 3: Elliptically polarized light (green) as a sum of lcp (blue) and rcp (red) light with different amplitudes. Viewed in –z direction (from the receiver to the light source). θ is the ellipticity.
2.2. ORD AND CD SPECTROSCOPY
In an optically active medium the two circularly polarized waves, into which a plane polarized light can be decomposed, will travel with different velocities and, as a result, the plane polarized wave will be rotated about the direction of propa-gation relative to the initial plane of polarization. The Optical Rotatory Dispersion (ORD) spectroscopy measures this rota-tion of polarization plane F = pnl(n- – n+)/c, where l is the light path length. When in an optically active medium the absorp-tion bands exist then the absorption of lcp and rcp light is different and a Circular Dichroism (CD) signal DA = Alcp - Arcp appears. The CD signal can be also presented as a difference of molar extinction coeffi cients Dε = εlcp - εrcp. The relationship between absorption and molar extinction coeffi cient is given by the Beer-Lambert law A = e C l with C being the molar con-centration of absorbing molecules. Usually the results of CD measurements are presented in DA for the cases when the concentration is unknown and in Dε for a known concentra-tion. In spectropolarimeters the signal is plotted as the elliptic-ity q of the light after passing through the sample (the tangent of this value is the ratio of the minor axis to the major axis of the ellipse, Fig. 3). With units in radians it is connected with the difference in absorption as q = 0.5757DA. In millidegree units the molar ellipticity is given by [q]M = 3298.2 De.
ORD and CD in materials with absorption bands are closely re-lated to each other. The results of both experiments are con-nected via Kramers-Kronig transformations.
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In eq. (4) 2/)( yx immm ��� with m = Sei ri being the electric dipole moment operator (ri
is the position vector of the ith par-ticle with the charge ei), Lz and Sz are the z components of the total electronic orbital and spin angular momentum operators and da is the electronic degeneracy of state a. Symbols a and b label components of the electronic states a and b, respectively.
Figure 4 illustrates the appearance of the A-term for the sim-plest case of the transition from non-degenerate ground state to triply degenerate excited state. Magnetic fi eld splits the com-ponents of the excited state. As a result, left and right circularly polarized light is absorbed at different energies. Since the Zee-man splitting is much smaller than the linewidths, the opposite-ly signed transitions for left and right circularly polarized light al-most cancel each other leading to a derivative band shape. The corresponding signal is weak and temperature independent.
B-term appears due to magnetic fi eld induced mixing of dif-ferent states of the complex. Both ground and excited states can be affected, however, usually the contribution of the ex-cited state mixing is larger since the energy gap between cor-responding states is smaller. Neglecting ground state mixing, the expression for B0 parameter is
� � �� �
��
amkbmaEE
kSLbd
Bbk
zz
bka��
!
���
�� �2
Re2
,,
0
�� �� amkbma �� . (5)
Symbol k labels components of the electronic state k that is mixed with the excited state b by the applied magnetic fi eld. Owing to this mixing, the absorption for left and right circularly polarized light becomes slightly different. The resulting sig-nal has band shape, is weak and temperature independent (Fig. 5). If B-term in the a ® b absorption appears due to the admixture of state k to state b, the B-term in the a ® k absorp-tion due to the admixture of state b to state k is also present in the spectrum, both terms being of opposite signs and shifted in energy. In the case when energy separation of these terms is small they can overlap giving rise to a “pseudo” A-term.
C-term appears when the ground state is degenerate. Magnetic fi eld splits this ground state. Like in the case of A-term it leads to the absorption of the left and right circularly polarized light at different energies. However, the most signifi cant effect is the difference in thermal population of the components of the ground state. The oppositely signed transitions for left and right circularly polarized light do not cancel each other. The resulting signal is strong and has the band shape (Fig. 6). The intensity of this signal depends on temperature and increases with the temperature decrease. The corresponding C0 parameter is
� �22
,0 2
1 ��������
bmabmaaSLad
C zza
�� ���� � . (6)
It should be emphasized that the matrix elements of m+ (m-) op-erators are associated with the absorption of the lcp (rcp) light, respectively. So, the corresponding selection rules are DS = 0 and DL = ±1 with DML = +1 for the left circularly polarized light and DML = -1 for the right circularly polarized light.
Fig. 4: Illustration of the origin of the MCD A-term: corresponding optical transitions for the left and right circularly polarized light (left panel); absorp-tion profi le of the lcp and rcp light (right panel, top) and resulting MCD signal (right panel, bottom).
Fig. 5: Illustration of the origin of the MCD B-term: corresponding optical transitions for the left and right circularly polarized light (left panel); absorp-tion profi le of the lcp and rcp light (right panel, top) and resulting MCD signal (right panel, bottom).
Fig. 6: Illustration of the origin of the MCD C-term: corresponding optical transitions for the left and right circularly polarized light (left panel); absorp-tion profi le of the lcp and rcp light (right panel, top) and resulting MCD signal (right panel, bottom).
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As can be seen from eq. (3), in the linear limit (when Zeeman splitting is small compared to kBT and to bandwidth of the ab-sorption line) the contributions of A, B and C terms are addi-tive. The A term can be separated via its derivative band shape while B and C terms can be easily distinguished due to the temperature dependence of the C term. The relative intensities of all 3 terms can be estimated as
TkEE
CBABbak
1:
1:
1::
)(�"� (7)
Typical line width of a broad band is about 1000 cm-1. Assum-ing that Ek – Ea(b) is about 10000 cm-1 the relative intensities at room temperature (kBT » 200 cm-1) can be estimated as A:B:C = 10:1:50. Going down to the liquid helium temperature the magnitude of C-term increases as 1/T, so for the paramag-netic ions at low temperatures this term is dominant.
One more value that is used during the analysis of MCD data is the dipole strength of the absorption intensity in the absence of an applied magnetic fi eld:
� �22
,0 2
1 ������
bmabmad
Da
�� �� � . (8)
Using this value in combination with A1 and C0 parameters it is possible to obtain the information about the magnetic proper-ties of the excited and ground states. For the example given in Fig. 4 one derives that in the linear limit A1/D0 = 2gex, where gex is the g-factor of the triply degenerate excited state. Similarly, for the example given in Fig. 6 one fi nds C0/D0 = 2ggr with ggr being the g-factor of the triply degenerate ground state [7].
If several low lying states are thermally accessible and the Zeeman interaction between them is small compared to the energy separation, MCD spectrum includes the additive con-tributions of individual transitions from each of these states (eq. (3)) multiplied by the corresponding population factors [7].
Equations (4)-(6) represent the A1, B0 and C0 parameters only for the case when the laboratory and molecular coordinate systems coincide. Usually, the MCD experiment is performed on glasses, mulls or solutions, which contain randomly oriented molecules. The orientationally averaged 1A , 0B and 0C parameters can be found in [7]. For the molecules with high symmetry the proce-dure of the calculation of these parameters is simplifi ed with the use of Wigner-Eckart theorem, the irreducible tensor method and properties of the coupling coeffi cients for the correspond-ing symmetry point group. A complete description of used tech-niques is presented in the book by Piepho and Schatz [7].
2.3.2 VTVH-MCD EXPERIMENT – APPARATUS, SAMPLE PREPARATION AND PRESENTATION OF RESULTS
For MCD measurements a spectropolarimeter, the same as for CD measurements is needed. The accessible wavelength range should be between 200 nm and 800 nm, better up to 2000 nm, Near-Infrared (NIR) region. For low temperature measurements a cryostat is needed.
The superconducting magnet with the magnetic fi eld oriented parallel to the lcp/rcp light beam should allow measurements up to 8 T or higher. Use of an electromagnet would limit the fi eld strength to about 1.5 T. The cryostat should provide low temperatures around the sample of 2 K or below. It is advisable to take spectra with the magnetic fi eld parallel and antiparallel to the travelling light allowing to extract contributions from glass imperfections or natural CD effects. The principal scheme of the experiment is presented in Fig. 7. The photo of our experi-mental setup is shown in Fig. 8.
Fig. 8: Magnetic Circular Dichroism experimental setup. On the fi rst plan a spectropolarimeter is seen under the table, nearby is a small pump with a fl ow regulator to pump helium vapor, further a helium cryostat is seen, which is con-nected with the spectropolarimeter by the light tube. Right to the cryostat there is a light-tight box with a light detector and quartz lenses mounted on the opti-cal bench. Electronic panel with a temperature controller, liquid helium monitor and superconducting magnet supply are behind the table and not seen.
Fig. 7: Schematic illustration of the MCD experiment. lcp and rcp light beams are switched on alternatively.
MCD signal calibration may be checked using aqueous solu-tion of CoSO4·7H2O, 0.05 M, for which molar magnetic ellipticity [q]M,510 nm equals to -6.2·10-3 deg·cm2·dmol-1·G-1 [8]. The correspond-ing equation is q = 0.01 [q]M·l·C·B, where q is the directly meas-ured ellipticity in degrees, l is the path length in cm, B is magnetic induction in G and C is the molar concentration in dmol/cm3.
MCD is usually measured as ellipticity being related to the dif-ference of absorption of lcp/rcp light. It is recommended to measure simultaneously MCD and absorption spectra. In princi-ple, the preparation of samples for MCD and absorption meas-
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urements is similar. As always, it is the best to have a single crystal. If an accordingly thin transparent crystal is not obtained by crystallization, it has to be prepared by cutting and polishing.
When a single crystal is not accessible or the substance is amorphous, then samples can be prepared for measurement by dissolving in a proper solvent. The solubility must be suffi cient and the absorption behavior of the solvent must be known. The structure of the molecule must not be changed via solving when compared with solid state properties. When this is not assured or solid properties are important, as for the case, e.g., of magneti-cally ordered materials, then spectra are mostly taken from solid state microparticles prepared in a Nujol as a mull or pressed with the KBr buffer into pellet. Not always Nujol is a good glass-ing agent, sometimes glycerol, ethylene glycol, tetramethylsilane or some other are better. For nanoparticles the colloidal disper-sions in liquids or grafted on quartz plates may be used.
Liquid or mull samples are usually placed between thin plates made from melted quartz. During freezing they go to a glassy state. Cracks, in particular, by the use of polymers, can lead to scattering effects. The depolarization of incident light might lead to not accurate data. The optimum amount/concentra-tion of the sample must be evaluated before starting the fi nal measurements. In principle, for both, the MCD and the absorp-tion spectra, the polarization properties of the solid species should be known and if possible the crystal structure within the temperature of interest.
MCD measurement data usually are presented in two ways: MCD spectrum (signal intensity as a function of wavelength or energy) and saturation magnetization curves (signal intensity at some fi xed wavelength as a function of μBB/2kBT). Analysis of MCD spectrum allows to determine the energies of the cor-responding transitions while the behavior of the magnetization curves gives the information about the magnetic properties of the ground state. For example, for an isolated Kramers doublet with an isotropic g-value the MCD signal behaves as
� � � �TkBg BB 2/tanhmax �-- � . (9)
As a result, the ground state g-value can be determined from a plot of the MCD intensity vs. μBB/2kBT at fi xed wavelength.
2.4. COMPARISON OF ELECTRONIC ABSORPTION AND MCD SPECTRA
Both MCD and absorption spectra are composed of the optical bands corresponding to the individual transitions. It is assumed that these individual bands are Gaussian in shape. Although MCD spectra seem to be more useful than absorption spectra it is the best to measure both. Then by a simultaneous Gaus-sian analysis of both spectra the individual transitions can be resolved. Here we would like to demonstrate one example.
The absorption and MCD spectra of tetrameric (C10H22NOCuCl)4 complex [9] dissolved in ethanol are shown in Fig. 9. It is seen that the use of the MCD technique leads to the multiplication of the infor-mation compared to the absorption spectrum. Due to different sig-
nal signs, more transitions (≥ 11) can be differentiated in the MCD spectrum, whereas in the absorption spectrum only several broad bands are visible. The symmetry of the nearest surrounding of each Cu(II) ion is very low [9]. As a result, four d-d transitions should be present in the corresponding region of both spectra. In the MCD spectra all of these d-d transitions are seen (three distinct lines and a shoulder) whereas the absorption spectrum in this spectroscopic region represents one structureless line. Additionally, the intensi-ties of the d-d transitions are comparable with the charge-transfer ones in the MCD spectrum, while in the absorption spectrum they are very weak (see inset to Fig. 9). The detailed assignment of the transitions in (C10H22NOCuCl)4 complex is in progress [9].
Usually the assignment of d-d transitions can be done with the ligand fi eld theory. In last years the assignment of all transitions (d-d and charge transfer) in the absorption and MCD spectra is based on Density Functional Theory (DFT) calculations. As an ex-ample we refer to [10] where the molybdenum(V) complex [Mo(O)Cl3dppe] was investigated with the use of UV-VIS and MCD spec-troscopy. The characteristic feature of the MCD spectrum, mainly composed of C terms, is a presence of two “pseudo” A-terms centered at 308 nm and 370 nm. An assignment of the observed transitions was performed on the base of the time-dependent DFT calculations. The MCD signs of the individual transitions were determined using molecular orbitals obtained by DFT. The observed two “pseudo” A-terms were assigned to the ligand to metal charge transfer transitions from the pp orbitals of the equa-torial chloride ligands to the dxz and dyz orbitals of molybdenum.
For practical use in the assignment of MCD spectral lines the powerful tool is the program package ORCA by F. Neese [11].
Fig. 9: Absorption spectrum (top) and MCD spectrum (bottom) of tetrameric (C10H22NOCuCl)4 complex [9]. Inset is the part of the absorption spectrum corresponding to d-d transitions.
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3. CHARACTERIZATION OF THE MAGNETIC GROUND STATE
3.1. MAGNETIC INTERACTIONS
We are focusing on the dn -ions in the case when the spin-orbit interaction is relatively small compared with Coulomb interac-tion between electrons in atom. In the spherical symmetry the atomic many electron terms can be labeled as 2S+1L, where S and L are values of the total spin and orbital angular momen-tum, respectively. The Zeeman Hamiltonian for 2S+1L term is
� �BSLH eBZe g�� �ˆ , (10)
where μB is the Bohr magneton and ge is the Lande factor for a free electron. In eq. (10) L and S are operators of the total orbital angular momentum and total spin, respectively.
The spin-orbit interaction in general case can be written as
��i
iiiSO slH #ˆ , (11)
where li, si and ζi are the operator of orbital angular momen-tum, spin operator and spin-orbit coupling constant for the ith electron, respectively. For the 2S+1L term this equation can be rewritten as
LSH SSO $�ˆ , (12)
where SS 2/#$ �� with the plus sign for the less than half-fi eld shell and the minus sign for the more than half-fi eld one.
In the cubic ligand environment the 2S+1L term is split into 2S+1Γ terms with Γ being A1, A2, E, T1 or T2. The nonzero matrix ele-ments of the orbital angular momentum are only within the cubic orbital triplets T1 and T2. To calculate these matrix ele-ments, the proportionality of the matrix elements of the orbital angular momentum within the cubic orbital triplet and within the p-basis can be used (so called, T-P isomorphism) [12]. As a result, the matrix elements of the orbital angular momentum within T1 or T2 triplet can be calculated using the fi ctitious orbit-al angular momentum L=1 with the corresponding proportion-ality factor that depends on the case under examination. For example, the ground state of the high spin cobalt(II) ion in the octahedral surrounding is 4T1g orbital triplet. Matrix elements of the spin-orbit interaction within this term can be calculated with the use of the following operator:
LSH $2
3ˆ ��SO , (13)
where λ = -170 cm-1 and κ is the orbital reduction factor. This orbital reduction factor takes into account not only the covalen-cy of the cobalt–ligand bonds but also the mixture of both 4T1g states (originating from 4F and 4P atomic terms) by cubic crys-tal fi eld. Neglecting covalency, we fi nd that κ varies between 1 (weak cubic fi eld limit) and 2/3 (strong cubic fi eld limit).
The low-symmetry (non cubic) distortion of the ligand environ-ment within the T1 or T2 triplet can be also taken into account with the use of this fi ctitious angular momentum L=1:
� � � �222 3/2ˆyxzCr LLVL ��� �H , (14)
where D and V are the axial and rhombic parameters, respec-tively, of the tensor that accounts for non cubic distortion of the local surrounding. In eq. (14) all operators are written in the coordinate system where this tensor is diagonal. The Zeeman part of the Hamiltonian eq. (10) for the T1 or T2 triplet is also calculated with the use of the fi ctitious angular momentum L=1 taken with the corresponding coeffi cient.
If the ground 2S+1Γ term is orbitally non-degenerate (due to electronic confi guration or strong distortion from the cubic surrounding), the orbital angular momentum disappears. The magnetic behavior of this kind of complex can be described with the use of the Zero Field Splitting (ZFS) Hamiltonian
� � � �222 3/)1(ˆyxzZFS SSESSSD �����H , (15)
where D and E are the axial and rhombic parameters of the ZFS-tensor, respectively. As an example, Fig. 10 demonstrates the effect of the different ZFS parameters on the ground term of the S = 1 spin-only system. In the perfect octahedral surrounding (both parameters in eq. (15) are zero) this ground term is 3-fold degenerate in MS values (-1, 0, 1). In the axial environment this term is split into the singlet (MS = 0) and doublet (MS = ±1). A rhombic distortion leads to the further splitting of the doublet in two components with the energy gap between them being 2E.
Fig. 10: Energy splitting of the S = 1 sublevels by axial and rhombic distortions.
If the investigated complex contains more than one magnetic center, the exchange interaction between these ions can ex-ist. If the ground states of both interacting ions are orbitally non-degenerate the exchange interaction between them can be written using a spin-only Hamiltonian (eq. 16)
� �2121212ˆ SSdDSSSSH %���� exex J , (16)
where the fi rst, the second and the third terms represent the isotropic, anisotropic and antisymmetric contributions, respec-tively, with Jex, D and d being the parameters of the correspond-ing interactions. The parameter of the isotropic exchange is a scalar while the parameters of the anisotropic and antisym-metric terms represent a traceless symmetric tensor and an anti symmetric vector, respectively.
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The isotropic exchange interaction leads to the parallel (fer-romagnetic exchange) or antiparallel (antiferromagnetic ex-change) alignment of spins of the interacting ions. The ani-sotropic part of the exchange between ions with the orbitally non-degenerate ground states represents some correction to the isotropic one. The relative orientation of the interacting spins is still determined by the isotropic part of eq. (16), how-ever, the directions in space become non-equivalent and some preferable direction for spin alignment appears. This effect combined with the local anisotropy results in the anisotropy of the exchange coupled system as a whole. Finally, the antisym-metric exchange interaction leads to some inclination of the interacting spins with respect to each other.
Since for orbitally non-degenerate ions the contribution of the isotropic part of the exchange interaction is the dominant one, in many cases for the explanation of the powder averaged magnetic behavior of clusters composed of this type of ions, the Hamiltonian includes only the isotropic exchange interac-tion and ZFS part in the form of eq. (15).
If the exchange interaction takes place between ions with the orbitally degenerate ground states, in general, the orbital contributions in the exchange Hamiltonian should be present [13]. However, in many cases the magnetic behavior of these complexes can be explained with the use of the isotropic ex-change interaction taking place between real spins of interact-ing ions [14]. The anisotropy of the magnetic behavior of the corresponding complexes due to the orbital contributions are included in the Hamiltonian as spin-orbit interaction and low-symmetry crystal fi eld terms eq. (14) [15].
3.2. MCD C-TERM OF MONONUCLEAR COMPLEXES
The general approach for the analysis of the C-term behavior was presented in [16]. Two cases are distinguished, namely, orbitally degenerate and orbitally non-degenerate ground state a. In the fi rst case the MCD signal appears in the zero-th order. Using Wigner-Eckart theorem, the irreducible tensor method and properties of the coupling coeffi cients for symmetry point groups, authors demonstrate that for the mononuclear com-plex with the orbitally degenerate ground state the MCD C-term behavior can be calculated as:
Mxy LzT + Myz LxT + Mzx LyTΔeE
, (17)
where Mij are some parameters that include the matrix ele-ments of i-th and j-th components of the electric dipole opera-tor [16] and
TkL is the thermally and orientationally averaged k-th component of the orbital angular momentum within the ground state:
� ���
�
�
ZTkE
LL BkTk
/exp
4
1 �))�
�
'(( ddAzk sin
*
. k = x, y, z (18)
In eq. (18) Z means the partition function and '( cossin�zxA ,'( sinsin�zyA , (cos�zzA with angles ϑ and ϕ describing
the orientation of the z-axis of the laboratory-fi xed coordinate system (and hence the external magnetic fi eld) with respect to the molecule-fi xed frame of reference. The summation is performed over all sublevels a of the ground state a. The cor-responding wave functions and the energies Ea depend on the spin-orbit interaction within this ground state as well as on the strength and orientation of the applied magnetic fi eld.
For the complexes with the orbitally non-degenerate ground state the MCD signal appears as a result of spin–orbit admix-ture of some states to the ground and/or excited states. To ob-tain equation for the C-term behavior in the case of the mono-nuclear complex with the orbitally non-degenerate ground state one should replace in eqs. (17), (18) components of the orbital angular momentum with the corresponding compo-nents of spin of the complex. Mij in this case differ from those in eq. (17) and, in addition to reduced matrix elements of the electric dipole moment operator, include also reduced matrix elements of the spin-orbit coupling [16,17].
For the detailed simulation of the MCD spectrum and the analysis of the transitions one needs to calculate Mij values in eq. (17) explicitly. However, if only the magnetic behavior of the ground state is analyzed, these values can be regarded as some parameters during the simulation of the saturation mag-netization curves.
TkL (or TkS )-values can be calculated
with the use of the Hamiltonian that includes magnetic interac-tions presented in section 3.1.
3.3. MCD C-TERM OF THE EXCHANGE COUPLED CLUSTERS
Exchange interaction between ions with nonzero spin values affects the magnetic properties of the whole cluster and, as a consequence, the temperature and magnetic fi eld behavior of the MCD signal. Assuming that the exchange is weak enough and can be regarded as perturbation acting within the direct product of the wave functions of non-interacting ions, one obtains that the MCD signal of the exchange coupled cluster consists of contributions coming from each of interacting ions [18,19]. Temperature and magnetic fi eld behavior of each con-tribution can be simulated with the use of eqs. (17), (18) (with Lk or Sk for the orbitally degenerate or non-degenerate ions, respectively), however, the averaging is now performed within the ground state of the whole exchange coupled cluster. The ground state magnetic properties of the entire complex strong-ly depend on the value of the exchange coupling parameter and this is refl ected in the MCD spectra via the temperature and magnetic fi eld dependence of the
TkL and TkS -values.
As it was already pointed out, the powder averaged magnetic susceptibility and magnetization behavior of the exchange cou-pled clusters can be often explained neglecting the anisotropy of the exchange interaction. Since the behavior of the MCD signal depends on the polarization of the corresponding transi-tions, even for a frozen solution the orientation selectivity is provided. So, it is expected that MCD studies can give some more information about the anisotropy of the exchange inter-action. The Mij-parameters for the exchange coupled clusters
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were found to be the same as in the case of no exchange. It means that the weak exchange does not change the optical properties of the interacting ions. The position of optical lines and the polarization of the corresponding transitions are the same as without exchange interaction [19].
In the case of a strong exchange interaction the contributions of the individual ions to the MCD signal of the whole cluster can not be differentiated. For clusters where strong exchange couples orbitally non-degenerate ions, the analysis of the MCD behavior can be performed with the use of eq. (17) for mono-nuclear complex with
TkS being the net expectation value of the total spin of the entire cluster [20]. The corresponding equation for the case of orbitally degenerate ions, to the best of our knowledge, was not presented in the literature.
3.4. MEASUREMENTS OF THE MAGNETIC SUSCEPTIBILITY AND THE MAGNETIZATION
Nowadays, the apparatus used to measure magnetic prop-erties is a Superconducting Quantum Interference Device (SQUID) magnetometer or a Vibrating Sample Magnetometer (VSM). With these apparatus the measured quantity is the magnetic moment or the magnetization M. Susceptibility is obtained as the M/H ratio with H being the strength of the applied magnetic fi eld. The SQUID magnetometer is most sen-sitive. Faraday balance measures susceptibility but is nowa-days rarely used because is less versatile. In particular, it is not suited for precise magnetization measurements and not at all for measurements of dynamic properties. With the SQUID mag-netometer or the VSM the magnetic moment may be measured as a function of temperature, magnetic fi eld, frequency, time and the sample orientation with respect to the fi eld direction. Except of direct current (DC) measurements an alternating cur-rent (AC) option presents access to study of dynamic magnetic properties. Commercial magnetometers allow measurements in the temperature range 1.7 – 800 K and in fi elds up to 7 T (SQUID) or 18 T (VSM). In Dresden High Magnetic Field Labo-ratory, magnetization measurements are performed in pulse fi elds up to 60 T [21]. Measurements in fi elds above 100 T are possible only with the Faraday rotation method [22].
The data obtained from experiment are used to fi t a suitable model. The values of the magnetization and the magnetic sus-ceptibility for an arbitrary direction (ϑ,ϕ) of the applied mag-netic fi eld can be calculated as
� � � �� �* +� �'(
'('(,
,ln,
BBZTkNM BA �
�� , (19)
� � � �� �'(
'('(,,
,,
HM
� , (20)
where Z is the partition function and kB and NA are the Boltzmann’s constant and Avogadro’s number, respectively. Frequently, poly-crystalline materials are measured, then by fi tting a suitable model the powder averaging must be done. The powder averaged magnet-ic susceptibility can be calculated simply as χav = (χx + χy + χz)/3, whereas for the calculation of magnetization the averaging over all
possible orientations of the external magnetic fi eld must be per-formed. This overall procedure may be diffi cult when clusters with several paramagnetic exchange coupled ions are treated. Then it may be necessary to use some simplifi cations, restrictions or in-voke some more advanced theoretical methods.
Interested readers may fi nd some good introduction to mag-netic measurements of various class of materials in [23].
Below, we show two examples of magnetic data collected with a SQUID magnetometer for powder samples in fi eld of 0.1 T.
The fi rst example is for antiferromagnetically coupled cop-per dimer. It is a (μ-phenoxo)-(μ-hydroxo)dicopper complex of chemical formula [Cu2(μ-OH)(L2)-(ClO4)]ClO4 (L2=C27H33ON6). Its molecular structure is shown in Fig. 11.
Magnetic data for this compound are presented in Fig. 12.
Fig. 11: Molecular structure of [Cu2(μ-OH)(L2)-(ClO4)]ClO4. Reproduced from [24] with permission.
Fig. 12: The magnetic data for [Cu2(μ-OH)(L2)-(ClO4)]ClO4. Blue line is the mo-lar susceptibility and red line is the effective magnetic moment (see text). Solid lines are fi ts. Adapted from [24] with permission.
As seen, susceptibility χ fi rst increases with decreasing tem-perature, shows maximum at about 90 K, then decreases down to 14 K and increases again a little. This last increase is due to monomeric paramagnetic impurities. A convenient form of
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presentation is to show the temperature dependence of χT. The case of χT =const corresponds to paramagnetism, χT de-creasing with decreasing temperature corresponds to antifer-romagnetism and χT increasing with decreasing temperature corresponds to ferromagnetism. However, with this form of pres-entation the contribution of impurities is not seen. Therefore, it is well to show sometimes both χ(T) and χT vs. T dependences.
The second example is for a ferromagnetically coupled dimer [Cu2(1,3-tpbd)(H2O)2(ClO4)3]ClO4 (1,3-tpbd = 1,3-bis[bis(2-pyri-dylmethyl)amino]benzene). Its molecular structure is present-ed in Fig. 13.
Assuming g1 = g2 = g and isotropic exchange interaction the following simple formula may be derived
� �� �TkJTkgN
BexB
BA
/2exp3
2 22
���
�,
2NA . (22)
This is the well known Bleaney-Bowers formula [26], applicable for low fi elds. It describes both antiferromagnetic (Jex < 0) and ferromagnetic (Jex > 0) interactions. It is often completed with paramagnetic contribution of monomer copper ions as well as with copper diamagnetism contribution and temperature inde-pendent paramagnetism (TIP).
For the fi rst compound the following best fi t parameters were obtained: Jex = -50 ± 2 cm-1, g = 2.00 ± 0.03, molar percent of impurities = 0.4%, TIP = 0.549 · 10-3 emu/mol. Solid lines in Fig. 12 are the best fi t lines.
For the second compound in addition to the exchange and Zee-man interactions authors included in the model the intermo-lecular interaction in the form of the Curie-Weiss law with the Weiss constant Θ for the whole dimer (for details see [25]). The best fi t parameters are: Jex = +4.6 cm-1, g = 2.12, Θ = -0.08 K. Solid lines in Fig. 14 are the best fi t lines.
3.5. ZERO FIELD SPLITTING; WHAT IS THE BEST METHOD TO DETERMINE IT.
Let us regard the ion in which the 2S+1Γ ground state has no fi rst order orbital angular momentum. Zero Field Splitting is the splitting of 2S + 1 spin degeneracy of this ground multi-plet (S>1/2) in the absence of the applied magnetic fi eld. This splitting takes place mainly due to the spin-orbit coupling with the excited states combined with the low-symmetry (non cubic) crystal fi eld of the ligand surrounding. For the ions with even number of unpaired electrons the ZFS can remove the spin de-generacy completely (see Fig. 10) while for those ones with odd number of unpaired electrons all levels remain doubly degen-erate (Kramers degeneracy).
The axial D and rhombic E ZFS parameters have been already introduced via eq. (15). The knowledge of the sign and the val-ues of these parameters are important for the description of the properties of the complexes under study. For example, in the Single Molecule Magnets (SMMs) D is related to the height of the barrier for magnetization reversal, see chapter 4.4.
The absolute value of D-parameter can vary from several to about 100 cm-1 and higher. There are several methods for evaluating ZFS parameters, namely, Electron Paramagnetic Resonance (EPR) spectroscopy, magnetic measurements, MCD, Far Infra-red spectroscopy, Mössbauer spectroscopy, Inelastic Neutron Scattering, calorimetry etc. Depending on the absolute value of D-parameter different experimental methods can be used. The X-band EPR method provides very accurate values of the ZFS parameters in the case when the energy splitting is not very high. For higher values of ZFS parameters the information can be ob-tained with the help of High Field EPR due to much larger Zee-man splittings (currently at some laboratories equipments up to
Magnetic data are presented in Fig. 14 in the form of χT vs. T de-pendence. As seen, χT initially grows with decreasing tempera-ture, then shows a maximum at 3.5 K. The decrease below 3.5 K is due to intermolecular antiferromagnetic interactions. The ini-tial increase points to the ferromagnetic intramolecular coupling.
Fig. 13: Schematic representation of the molecular structure of the cation of [Cu2(1,3-tpbd)(H2O)2(ClO4)3]ClO4. Reproduced from [25] with permission.
Fig. 14: Thermal variation of χT of [Cu2(1,3-tpbd)(H2O)2(ClO4)3]ClO4. The inset shows the maximum in the low temperature region. The solid line is the best fi t. Reproduced from [25] with permission.
The Hamiltonian for two interacting ions with spin 1/2 as for d9 systems can be written in the following form
� �BSSSSH 2211212ˆ ggJ Bex ���� � . (21)
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360 GHz and up to 30 T are in use or in perspective [27,28]). However, still data at about 100 cm-1, as reported [29] for some Co(II) complexes, are not accessible using this method. In this case the information can be obtained by the analysis of magnet-ic susceptibility and magnetization measurements as well as by the VTVH-MCD experiments. VTVH-MCD technique is very sensi-tive and allows the measurements of a small amount of sample, however, it requires the sample to be transparent. Very recently Boča et al. [29] reported on Far Infrared experiments allowing the detection of ZFS values in the order of 100 cm-1.
An overview of different methods and the so far determined D-values is given in [30]. It is concluded that each of the used techniques has some limitations and can not be declared as the best one in the determination of the D-values. A researcher should view these methods as complementary ones.
4. SELECTED EXAMPLES
4.1. DETERMINATION OF THE LIGAND FIELD ENERGIES IN DIAMAGNETIC ANION PtCl4
2-
An example of the use of A-terms in the assignments of ligand fi eld transitions can be found in [31] where the MCD study of PtCl4
2- was described. Pure d-d transitions are parity forbidden, however, they become allowed due to the admixture of odd parity functions with the same symmetry into the d-orbitals. This ad-mixture takes place due to vibrations (in centrosymmetric com-plexes) or static distortions (in low symmetry ones). The sche-matic energy diagram for d-orbitals in a square-planar complex is shown in Fig. 15, however, the order of low lying occupied orbitals depends on the ligands. In the square-planar environment Pt2+ ion (5d8) has 1A1g diamagnetic ground state. There are three d-d transitions in this complex into the empty 225 yxd
� orbital, namely, 1A1g→
1B1g ( 2zd → 22 yxd� ), 1A1g→
1Eg (dzx,dyz→ 22 yxd� ) and 1A1g→
1A2g (dxy→ 22 yxd
� ). The MCD spectrum in this area consists of a nega-tive B-term at about 25500 cm-1, a negative A-term at 30300 cm-1 and a positive B-term at 37600 cm-1. The band at 25500 cm-1 was assigned to 1A1g→
1A2g (dxy→ 22 yxd� ) transition based
on the polarization properties. The band at 30300 cm-1 was as-signed to the 1A1g→
1Eg (dzx,dyz→ 22 yxd� ) transitions since in this
complex the ground state is non-degenerate and A-term can ap-pear only for transitions to the degenerate excited state (see eq. (4)). Finally, the only possibility for the band at 37600 cm-1 is to correspond to 1A1g→
1B1g ( 2zd → 22 yxd� ) transition. This interpreta-
tion of the MCD spectra confi rms the energy ordering of the occu-pied 5d-orbitals in the investigated complex presented in Fig. 15.
4.2. VTVH-MCD STUDY OF IRON PROTEINS: EXAMPLE FOR ZERO FIELD SPLITTING
MCD spectroscopy has found considerable use in bioinorganic chemistry for studying different biologically signifi cant transition metal complexes (see for example [5, 8, 15, 32-37]). The behav-ior of C-term as a function of temperature and applied magnetic fi eld was found to provide useful information about the site ge-ometry of metal ions. Let us regard a high spin Fe(II) ion (d 6 con-fi guration) in a distorted octahedral surrounding. This situation is met in different iron enzymes. Combined action of the spin orbit interaction and a low symmetry (non cubic) crystal fi eld results in the splitting of the 5T2g cubic term. The splitting and magnetic be-havior of the low-lying levels can be described with the use of the ZFS Hamiltonian (eq. (15)) with S=2. The order of low lying levels as well as their splitting strongly depends on the ligand surround-ing and can be used as a “fi ngerprint” to derive the information about the site geometry of metal ion. Since this splitting occurs for non-Kramers doublets, the corresponding complexes are EPR silent. Solomon and his group developed methods of ana-lyzing Variable Temperature-Variable Field (VTVH)-MCD data for systems with non-Kramers doublet ground states for obtaining the information about the ground energy levels [32, 33]. It was demonstrated for a model compound that it is possible to calcu-late values for the ligand fi eld splitting parameters (D and V in eq. (14)) from the analysis of the magnetization curves (see Fig. 16). The obtained information can give a molecular-level insight into the processes involving the corresponding enzymes.
Fig. 15: Schematic energy diagram for d8 complex in a square-planar envi-ronment with z being the four-fold axis.
Fig. 16: MCD saturation magnetization data for FeSiF6�6H2O complex. Adapt-ed with the permission from [32].
4.3. DETERMINATION OF COORDINATION NUMBER FOR HIGH-SPIN Co(II) IONS
The splitting of the low-lying energy levels of the high-spin Co(II) ion strongly depends on the number of ions in the nearest neighbor surrounding. In the octahedral crystal fi eld (6 near-est ions) the ground state of the high-spin Co(II) ion is the 4T1g orbital triplet. As it was already pointed out in Section 3.1, the matrix elements of the spin-orbit interaction within the ground 4T1g orbital triplet of the high-spin Co(II) ion can be calculated with the use of fi ctitious orbital angular momentum L=1 and eq. (13). Spin-orbit coupling splits the 4T1g term into three lev-els corresponding to J = 1/2, 3/2 and 5/2 with J being the quantum number of the fi ctitious total angular momentum J = L + S of high-spin Co(II) in the 4T1g term. Since the value of the spin-orbit coupling parameter l is negative, the ground state corresponds to J = 1/2 Kramers doublet. This Kramers
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doublet is well separated from the excited levels. The corre-sponding energy gaps are indicated in Fig. 17 with l being about 170 cm-1. As a result, at low temperature the effect of the excited states on the MCD C-term behavior can be neglect-ed. The account of the low-symmetry (non cubic) crystal fi eld does not change this conclusion. So, the C-term behavior of the high-spin Co(II) ion in octahedral surrounding can be simu-lated with the use of eq. (9). The signal intensity vs. B/T curves plotted at different temperatures coincide.
In the tetrahedral crystal fi eld (4 nearest ions) the ground state of the high-spin Co(II) ion is the 4A2 orbital singlet. Due to the spin-orbit interaction with the excited states this state is split (Fig. 17 right). The splitting can be characterized with the help of the ZFS Hamiltonian (eq. (15)) with the absolute value of the axial ZFS parameter D usually less than 10 cm-1. The corre-sponding C-term behavior exhibits strong divergence of the sat-uration magnetization curves plotted at different temperatures.
was estimated to be 10±5 cm-1 and this term was ascribed to the 5-coordinated cobalt site. This assignment was also sup-ported with the help of the angular overlap method.
4.4. MCD STUDY OF SINGLE MOLECULE MAGNETS (SMMs)
One of the most fascinating developments of the last decades in the fi eld of molecule-based magnetism is the discovery and characterization of Single Molecule Magnets. These molecules exhibit magnetic bistability due to the energy barriers for mag-netization reversal and potentially can be used for high-den-sity data storage and quantum computing. The phenomenon was discovered in the family of clusters of general formula [Mn12O12(O2CR)16(H2O)4] [38,39,40] (Fig. 19). The exchange interaction results in the ground state S=10. The value of the axial ZFS parameter for this ground state was found to be DS=10 ~ -0.5 cm-1 that gives the barrier for magnetization reversal of about 50 cm-1. Below 3K the system shows an extremely slow relaxation of the magnetization.
Fig. 17: Idealized schemes for ground-state splitting in 6-coordinated (left) and 4-coordinated (right) Co(II) complexes.
Co(II) ion with 5 nearest neighbors demonstrates both the lev-els splitting in the zero fi eld and the corresponding MCD behav-ior in between octahedral and tetrahedral cases. So, the energy level splitting and the temperature and magnetic fi eld behavior of the MCD signal of high-spin Co(II) ion can be regarded as an indicator of the number of nearest neighbors of cobalt ion.
This dependence of the MCD signal behavior on the number of the nearest neighbors was used in the study of Escherichia coli methionyl aminopeptidase [34]. By the analysis of the tem-perature and fi eld dependent behavior of two major peaks at 495 and 567 nm, it was possible to differentiate between six- and fi vefold coordinated Co(II)-ions. The C-term at 495 nm be-haves like a signal from the isolated Kramers doublet (eq. (9)) with g-factor about 4.45 (Fig. 18 left) and was attributed to the 6-coordinated cobalt. The C-term at 567 nm demonstrates a nested behavior (Fig. 18 right). The value of the ZFS parameter
Fig. 18: MCD saturation magnetization data for EcMetAP at 495 nm (left) and 567 nm (right). Adapted with the permission from [34].
Fig. 19: Schematic view of the core of the [Mn12O12(O2CR)16(H2O)4] cluster. Blue circles are Mn(IV), the green ones Mn(III) and the small atoms are oxygen. The arrows represent the ground state confi guration with S=10. Re-printed with the permission from [20].
The complex was studied with the help of MCD above and be-low the blocking temperature [20]. The study above the block-ing temperature results in the parameters similar to those obtained from the magnetic measurements. The investigation below the blocking temperature demonstrates hysteresis loops in the signal intensity (Fig. 20). The sign of the spectrum de-pends on the direction of the initial magnetic fi eld.
Fig. 20: Hysteresis loops of [Mn12O12(O2CR)16(H2O)4] (R=n-C14H29) in poly(methyl methacrylate) fi lms measured at 1.7K and at 21200 cm-1 (fi lled squares) and 19600 cm-1 (open circles). Adapted with the permission from [20].
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4.5. MCD OF EXCHANGE COUPLED DIMERS
4.5.1. PECULIARITIES OF THE MCD SIGNAL OF THE EXCHANGE COUPLED DIMERS
An example of the analysis of MCD C-term saturation behavior of the exchange coupled dimer can be found in [18]. Authors studied three isostructural dimeric complexes Cr(III)Zn(II) (1), Ga(III)Ni(II) (2) and Cr(III)Ni(II) (3). Since Zn(II) and Ga(III) are di-amagnetic, complexes 1 and 2 from the magnetic and MCD C-term points of view can be regarded as isolated Cr(III) and Ni(II) ions, respectively, while in complex 3 these ions are exchange coupled. Values of the exchange integral and ZFS parameters were obtained both from the analysis of the magnetic suscepti-bility data and from the MCD saturation magnetization curves. The exchange interaction was found to be antiferromagnetic.
Comparison of the MCD spectra of 2 and 3 demonstrates that the contributions (bands) of Ni(II) ion to the corresponding spectra are of the opposite sign. Complex 2 can be regarded as an isolated Ni(II) ion. In the applied external magnetic fi eld the magnetic moment of Ni(II) tends to align parallel to the applied magnetic fi eld with spin being directed antiparallel (since the electron charge is negative). In the exchange coupled complex 3 the behavior of both ions is determined by the competition be-tween the magnetic fi eld and the antiferromagnetic exchange coupling (Jex = -8.4cm-1). The ground state can be regarded as a state with spin of Cr(III) oriented mainly antiparallel to the magnetic fi eld, while the spin of Ni(II) is mainly parallel to this fi eld. One fi nds that the calculated
TkS -values for Ni(II) ion in the case of 2 and 3, and hence the corresponding MCD sig-nals, differs in sign. The antiferromagnetic exchange interac-tion between Ni(II) and Cr(III) ions changes the sign of the MCD bands coming from Ni(II) ion. The same effect was found dur-ing the magneto-optical study of isostructural Ga(III)Co(II) and Fe(III)Co(II) dimers [41,42]. The contributions of Co(II) ion to the MCD spectra of both complexes are of the opposite signs.
Another interesting peculiarity of the MCD spectra for complex 3 is that at high temperatures the bands corresponding to Ni(II) signal demonstrate sign inversion (the behavior of signal at 768 nm is shown in Fig. 21). As it was already pointed out,
due to the antiferromagnetic exchange interaction at low tem-peratures and small magnetic fi elds the magnetic moment of Ni(II) in the Cr(III)Ni(II) dimer is oriented mainly antiparallel to the applied magnetic fi eld. With the temperature and/or mag-netic fi eld increase the energies of the corresponding interac-tions become comparable that results in the reversion of the magnetic moment of Ni(II) ion and, as a consequence, in MCD signal sign inversion.
One fi nds that the sign inversion of the MCD signal should be typ-ical for the antiferromagnetically coupled dimers composed of ions with nonequivalent spins. Similar effect can be observed in the dimers with the identical ions in the case of strong contribu-tion to the magnetic behavior of the unquenched orbital angular momentum. As an example the antiferromagnetically coupled high spin Co(II) dimer can be regarded [19]. The theoretically simulated MCD signal behavior for the complex composed of two equivalent high-spin Co(II) ions in axially compressed oc-tahedral surrounding with perpendicular local anisotropy axes is shown in Fig. 22. The sign inversion for the (xz+yz)-polarized MCD signal with the fi eld and temperature increase is predicted.
Fig. 21: MCD saturation magnetization curves for complex 3. Adapted with the permission from [18].
Fig. 22: Simulated MCD saturation magnetization behavior for Co(II) dimer at κ1 = κ2 = 1, Jex = -3 cm-1, D1 = D2 = -700 cm-1 and perpendicular local axes. A: Mxy(1) = Mxy(2), all other Mij = 0. B: Myz(1) = Myz(2) = Mzx(1) = Mzx(2), all other Mij = 0. Adapted with the permission from [19].
4.5.2. MAGNETIC AND MCD STUDY OF THE INFLUENCE OF TERMINAL LIGANDS ON THE BEHAVIOR OF Co(II) DIMER
The magneto-optical investigation of fi ve Co(II) dimers of the general formula Co2(μ-H2O)(μ-OOC-R)2(OOC-R)2(H2O)2L2 (R = CCl3) with the same central part and different terminal ligands (L = iso-propyl alcohol; 1,4-dioxane; tetrahydrofuran; part of the 1,2-di-methoxyethane group; pivalate) was presented in [43]. It was found that change of terminal ligands has a small effect on the magnitude of the exchange interaction. Contrary, the change of terminal ligands strongly affects both the value of the low sym-metry (non cubic) crystal fi eld and the direction of the local ani-sotropy axes. Since Co(II) ion possesses the unquenched orbital angular momentum, its magnetic properties (and, as a conse-quence, the magnetic behavior of the whole dimer) strongly de-pend on the value and the sign of this low symmetry fi eld. As a result, the change of terminal ligands has a signifi cant effect both on the magnetic properties of Co(II) dimers and on the MCD
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saturation behavior. As can be seen from Fig. 23, at low tempera-tures and small values of the magnetic fi eld the MCD saturation curves demonstrate a variety of behavior from a slow growth to a sharp increase of the signal intensity (compare Fig. 23 B and C, curves for 3K).
Ju et al. [45] studied Cr3+-doped BaTiO3 nanocrystalline fi lms prepared from as-synthesized nanocrystals. Using combina-tion of MCD and magnetization measurements they observed that for a suffi ciently small size of nanocrystals ca. 7 nm and the concentration of the dopant ions ca. 5 %, the ferromagnet-ic ordering of the dopant ion spins appeared by partial retain-ing of ferroelectricity. This observation was explained by the formation of extended structural defects at the grain boundary interfaces of the fi lm, which played a mediating role in the mag-netic ordering.
Schimpf and Gamelin [46], using MCD spectroscopy, demon-strated for colloidal Mn2+ or Co2+ doped CdSe quantum dots the tuning and inversion of excitonic Zeeman splittings with de-creasing temperature in a small magnetic fi eld. Depending on temperature a sign inversion of the MCD signal allowed to dis-criminate between two contributions to Zeeman splitting, name-ly between intrinsic contribution dominating at room tempera-ture and exchange contribution dominating at low temperature, arising from sp-d coupling between delocalized charge carriers and localized spins of magnetic impurities. The exemplary MCD spectra of colloidal quantum dots are shown in Fig. 25.
Yao [47] studied the electronic structure of the thiolate capped gold clusters Au25(SG)18 and Au25(PET)18, where SG denote glu-tathione and PET 2-phenylethanethiolate. The MCD spectra were interpreted as composed of B terms, which means, that both excited as well as ground states of the Au25 “superatom” are nondegenerate.
Fig. 23: MCD spectra at 3K (top) and magnetization curves at 505 nm (bottom) of C14H22Cl12Co2O13·2C3H8O (A), C16H22Cl12Co2O13 (B) and C40H78Cl8Co2O17 (C). Sol-id lines represent theoretical curves. Adapted with the permission from [43].
4.5.3. EXCHANGE COUPLED Co(II) DIMER WITH AN UNUSUAL MCD SATURATION BEHAVIOR
A peculiar behavior of the MCD spectra was observed for the exchange-coupled [Co2(μ-H2O)(μ-OAc)2(OAc)2(tmen)2] complex (tmen = N,N,N’,N’-tetramethylethyl-enediamine) with a change in the magnetic fi eld [44]. With an increase in the magnetic fi eld strength the intensity of particular lines initially increases, then passes through the maxima, decreases, disappears and ap-pears again with the opposite sign (Fig. 24). This behavior was explained by the overlap of the electronic transitions at different but near wavelengths. Both the MCD magnetization curves and magnetization data obtained by a SQUID magnetometer are well reproduced using the Hamiltonian, which takes into account an orbital magnetic moment.
4.6. MCD FOR NANOMATERIALS
Although MCD method has better resolution compared to opti-cal absorption both methods are often used together because one complements other. Below, some examples of the use of both these methods are given in studies of nanoparticles prop-erties. Samples for these studies are usually prepared in the form of fi lms made by spin coating of colloidal suspensions on the quartz disks.
Fig. 24: Experimental MCD spectrum of [Co2(μ-H2O)(μ-OAc)2(OAc)2(tmen)2] at 3, 5 and 25K. Curves labels denote the magnetic fi eld in Tesla. Adapted with the permission from [44].
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5. X-RAY MCD (XMCD), X-RAY MAGNETIC LINEAR DICHRO-ISM (XMLD) AND MAGNETIC LINEAR DICHROISM (MLD)
XMCD is MCD in the X-rays energy range [6], which usually is between hundreds electron volts up to about 2000 eV. For XMCD experiments a synchrotron light source is needed. It of-fers a high degree of circular polarization by a brilliant intensity, when the beamline is equipped with an undulator (so called Elliptically Polarizing Undulator). This method has some advan-tage over optical MCD, as selectivity for elements (depending on the selected absorption edge), sensitivity to the oxidation state or a possibility to determine separately orbital and spin contribution to the magnetic moment. It is at all very sensi-tive, very small amount of sample can be measured, e.g. Gam-bardella et al. [48] measured single chain magnet properties of Co chains on the Pt substrate. These Co chains being apart 20.2 Å covered very small part of the Pt surface. Researchers observed magnetic hysteresis at 10 K. For the determination of the orbital μL and spin μS magnetic moments the so called sum rules were used (see also [49]). They state, that
mL = –2mB(A+B)nh /(3C), (23)
mS = –mB(A–2B)nh /C, (24)
where nh is the number of holes in the 3d level, A and B are in-tegrals of the dichroic signal under L3 and L2 edges, respective-ly and C is obtained from the integration of the non polarized X-ray absorption spectrum (XAS); see Fig. 26. When intensities of L3 and L2 lines are equal, μL disappears. It should be noted that sum rules fail in some cases, e.g., for L3 and L2 edges of rare earths [50]. Gambardella et al. found for Co nanowires μL = 0.68 μB, the value, which is 5 times greater in comparison with bulk Co (μL = 0.14 µB).
Fig. 25: Absorption and MCD spectra of colloidal (a) Cd0.995Mn0.005Se and (b) Cd0.995Co0.005Se quantum dots, collected at B = 0.63 T. The bold red and blue MCD spectra represent the highest and lowest temperatures, respectively. Reproduced from [46] with permission.
Fig. 26: Model X-ray near edge absorption spectrum taken for left (red line) and for right (black line) circular polarization – upper panel. XMCD spectrum obtained as a difference for left and right circular polarization – bottom panel.
Another spectacular example of XMCD possibilities comes from the work of Kimura et al. [51]. They measured XMCD spectra of Mn sub-monolayer (0.46) grown on Ni (100) substrate and showed that Mn layer was ferromagnetically coupled with the Ni substrate.
The principle of XMCD may be explained on the example of the L edge X-ray absorption measurements, usually undertak-en to probe 3d electron properties. L edge is obtained when the electron is excited from the spin-orbit split 2p3/2, 2p1/2
core levels to the empty d valence states, see Fig. 27. This ex-citation is a dipole transition with the spin conservation, i.e., ΔS = 0 and other selection rules are ΔL = ±1, ΔML = ±1. Spin up (spin down) photoelectrons can only be excited to the spin up (spin down) 3d empty states. In a magnetic material (or in magnetic fi eld) there is imbalance in the number of available empty states with a specifi c spin direction so the probabilities of excitation of spin up and spin down photoelectrons are dif-ferent. The spin and orbital momentum of these electrons are connected via the spin-orbital coupling. It can be shown that mainly those 2p electrons are excited of which orbital motion is in the same direction as the circular motion of the incident
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light so the transition probabilities for two polarizations will be different. Since 2p3/2, 2p1/2 states differ regarding the direction of the orbital momentum with respect to the spin direction, this difference in the absorption of lcp and rcp light will be opposite for L3 and L2 edges. The total intensity of two L3, L2 lines will be proportional to the number of holes in 3d orbitals.
pretation is more diffi cult, because more parameters are need-ed for the description. However, MLD has different selection rules, so, it may provide a complementary to MCD information.
Fig. 27: Diagram illustrating principle of XMCD. The vertical arrows in 2p states represent spins and the horizontal arrows represent orbital motion.
Fig. 28: Comparison of XAS, XMCD and XMLD spectra measured for a bcc (001) Fe at the L2,3 edge. Adapted with the permission from [52].
It should be noted that XMCD signal is proportional to the local magnetization M vector and it is well suited to study ferro- and ferri-magnetic materials but it is less useful for studies of anti-ferromagnets, where M vectors cancel out.
The new spectroscopic method X-ray Magnetic Linear Dichroism (XMLD) was developed quite recently, which is suited for antifer-romagnets, and its application is quickly growing. By XMLD the X-ray beam is directed perpendicularly to the magnetic fi eld. The spectrum is obtained as the differential absorption for two linear polarizations: parallel and perpendicular to the fi eld. XMLD sig-nal is quadratic in M and depends strongly (in contrast to XMCD) on the crystallographic orientation of the antiferromagnetic di-rection. It is used preferentially to study antiferromagnetic order in thin fi lms or on the surfaces. With this method one can de-termine the size, the direction, and anisotropy of the magnetic moment of the specifi c atom [52]. Exemplary XMLD spectrum is shown in Fig. 28 together with XAS and XMCD spectra for com-parison. As seen, the main difference to XMCD spectrum is that L3 and L2 lines on XMLD spectrum have the same sign.
With both XMCD and XMLD the imaging microscopy techniques were developed to study magnetic domains [53].
The optical counterpart of XMLD is MLD [6]. This method is rarely used and usually in combination with MCD. The MLD signal is an order of magnitude weaker than the MCD signal. The inter-
6. CONCLUDING REMARKS
This tutorial review presents the basic theory of the MCD ef-fect and a broad range of issues referring to this effect as a research tool. It was shown that MCD may be used not only to the identifi cation of electronic transitions in optical spectra but also to the study of magnetic properties. By these studies the support from other methods as magnetization measurements or EPR may be very helpful. For the assignment of the MCD spectral lines existent programming tools can help.
The role of the A, B, C terms appearing in the master formula has been explained as well as how they can be used for the in-terpretation of MCD spectra of mononuclear complexes. It was discussed how magnetic exchange infl uences the MCD spectra of dimeric complexes and how the useful information about mag-netic exchange and Zero Field Splitting can be obtained. Two cas-es have been differentiated: with and without orbital degeneracy. The discussion was illustrated with examples. Some peculiarities and unusual behavior of MCD spectra have been also shown.
MCD is one of the key methods for characterizing the structure and function of metalloenzyms and their biochemical and cata-lytic functionalities. Readers interested on this topic can fi nd a lot of useful information in recent publications, e.g., in numer-ous papers of the Solomon group ([32], [33]).
MCD is a very sensitive and powerful optical method which has been quite recently used for characterizing nanomaterials. However, MCD can be equally used to study magnetic proper-ties of isolated or ordered magnetic systems. Even unstable molecules can be studied.
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During the last two decades the XMCD method was a fast de-veloping fi eld of research. It has been also shortly described in this review and some examples of its use have been given.
ACKNOWLEDGEMENT
W.H. acknowledges support by DFG via Ha 782/85. Z.T. is grateful for the MNiSzW grant NN202103238 (Poland). S.O. acknowledges support of the Supreme Council for Science and Technological Development of the Republic of Moldova (Project 11.817.05.03A) and SCOPES grant IZ73ZO_128078/1. Authors are thankful to Professor M. Zeppezauer for his encouragement.
[5] Johnson, M.K. (2000) CD and MCD Spectroscopy in Physical Methods in Bioinorganic Chemistry, Que, L. (ed.), University Science Book, Sausalito, CA, p.233-285.
[6] Mason, W.R. (2007) A practical guide to magnetic circular dichroism spectroscopy, Wiley-Interscience.
[7] Piepho, S.B.; Schatz, P.N. (1983) Group Theory in Spectroscopy with Applications to Magnetic Circular Dichroism, Wiley-Interscience Publi-cations: New York.
[11] Neese, F. (2005) ORCA, an ab initio density functional and semiem-pirical program package, Max-Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr, Germany.
[12] Sugano, S.; Tanabe, Y.; Kamimura H. (1970) Multiplets of Transition-Metal Ions in Crystals; Academic Press: New York.
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