Optical absorption spectroscopy in geosciences: Part I: Basic concepts of crystal field theory

Published:January 01, 2004
Abstract
Chapters Chapter 3 and Chapter 4 deal with optical absorption spectroscopy and comprise two interrelated parts. Part I, the present chapter, is intended as an introduction for the beginners, e.g. undergraduate students of geosciences, in order to help them acquire an understanding of the basic theoretical principles, focussing on the crystal field (CF) concept. After a short introductory section including some “technical” information, the reader is guided step by step through the development of the qualitative principles of the CF theory (CFT), referring to several aspects important for geosciences. The necessary concepts and tools are briefly outlined, whereas references to a selection of relevant textbooks and publications are given for further reading. Concise tables and figures help to illustrate and summarise important topics. Examples of the actual spectra are provided, mainly concerning the “manyelectron systems” (of the firstrow transition ions, i.e., 3d^{2,3,7,8}), since they cover the full diversity of the crystal field based spectroscopic aspects.
Part II (Chapter 4 in this volume – Andrut et al., 2004) deals with the quantitative aspects of the crystal field theory and its applications. It highlights the power of semiempirical methods for the calculation of energy levels of the transition metal complexes with arbitrary low symmetry.
Scope of the optical absorption spectroscopy chapters (Part I and Part II)
Chapters Chapter 3 and Chapter 4 deal with optical absorption spectroscopy and comprise two interrelated parts. Part I, the present chapter, is intended as an introduction for the beginners, e.g. undergraduate students of geosciences, in order to help them acquire an understanding of the basic theoretical principles, focussing on the crystal field (CF) concept. After a short introductory section including some “technical” information, the reader is guided step by step through the development of the qualitative principles of the CF theory (CFT), referring to several aspects important for geosciences. The necessary concepts and tools are briefly outlined, whereas references to a selection of relevant textbooks and publications are given for further reading. Concise tables and figures help to illustrate and summarise important topics. Examples of the actual spectra are provided, mainly concerning the “manyelectron systems” (of the firstrow transition ions, i.e., 3d^{2,3,7,8}), since they cover the full diversity of the crystal field based spectroscopic aspects.
Part II (Chapter 4 in this volume – Andrut et al., 2004) deals with the quantitative aspects of the crystal field theory and its applications. It highlights the power of semiempirical methods for the calculation of energy levels of the transition metal complexes with arbitrary low symmetry. One specific method, namely, the “Superposition Model” (SPM) of crystal field, is dealt with in detail. By applying these methods, the oversimplified symmetry approximations (e.g. to cubic O_{h} symmetry) often used in the literature so far can be circumvented and the highest possible information content can be retrieved from the spectra. On the basis of selected, geoscientifically relevant examples, the role of the actual site symmetry and local geometry around a transition metal ion in determination of its spectroscopic properties is highlighted. The advantages of optical spectroscopic methods in the studies of the individual transition ion polyhedra in a crystal structure as compared to the diffraction methods, which provide “averaged” structural information, are demonstrated. As an example, the first detailed analysis of the local geometry of the Cr^{3+}O_{6} polyhedra in the structure of ruby, Al_{2}O_{3}:Cr^{3+}, is presented in Part II. Thus, it is shown that optical absorption spectroscopy in particular and the related spectroscopic techniques in general can provide indispensable and valuable information. These methods can be used in combination with other analytical and/or diffraction methods to study a given ionhost system. An extensive list of literature completes Part II.
The authors want to point out that, as a matter of course, a full coverage of all aspects of optical absorption spectroscopy is not possible (nor was it attempted) within the limited space available. Instead, we concentrate on the crystal field aspects bearing on interpretation of the polarised optical absorption spectra of single crystals containing transition metal ions, highlighting the capabilities of the relevant techniques as well as the information that can be acquired. Other pertinent methods and aspects are dealt with in details in the references provided.
Introduction
Optical spectroscopy is concerned with the qualitative and quantitative measurements of the absorption, reflection, and emission of light on single crystals or powder samples in the spectral range 40 000 to 4000 cm^{−1} (250 to 2500 nm). This energy range covers the nearultraviolet, visible, and part of the infrared portion of the electromagnetic spectrum, in which the transitions between electronic states occur (as well as vibrational overtones or combination modes). Note that this spectral range exceeds from above and below the part of the electromagnetic spectrum the human eye is able to detect (26 300 cm^{−1}–12 800 cm^{−1}, i.e., 380 nm–780 nm) and which is referred to as our colour vision range.
In general, the most comprehensive information can be obtained on appropriately oriented single crystals using polarised radiation (e.g. including pleochroism and quantitative analysis). Unfortunately, single crystals of sufficient size are not always available. Hence, often the investigations have to be performed on powder samples. In this case, it is only possible to retrieve qualitative information as well as bulkaveraged properties.
The optical spectroscopy area can be broadly divided into three subareas. (1) The measurement of absorption spectra in transmission mode is the most widely used method for transparent materials. (2) Reflection measurements have to be applied if the sample is either totally absorbing the electromagnetic radiation or cannot be further thinned for transmission measurements. Besides, reflection measurements are widely used in the field of remote sensing (e.g. weather forecast, environmental studies, ore deposits, composition of the surface of planets and of stellar dust etc.). (3) Various methods of emission spectroscopy are concerned with the emission of photons following an absorption process. As an example, the reader is referred to the chapter about luminescence by Nasdala et al. (2004a) in this book.
Optical absorption processes
Below we categorise the processes, which contribute to the absorption spectra of minerals in the optical spectral range. The approximate energy ranges in solids and the theory describing the respective process are given in square brackets, whereas the physical nature of the process is very briefly described.
Ligand–metal and metal–ligand charge transfer (LMCT, MLCT) [mostly > 30 000 cm^{−1}; molecular orbital theory (MO theory), selfconsistent field procedures (e.g. SCFXα method)]. Transitions between the energy levels described by the states predominantly centred at the oxygen ligands of coordination polyhedra and those predominantly centred at the central ion.
Metal–metal charge transfer (MMCT; also: intervalence charge transfer, IVCT) [24 000–9000 cm^{−1}; exchange theory]. Electrons hopping between transition metal cations with different valence in edge or facesharing coordination polyhedra.
Crystal field d–d transitions [40 000–4000 cm^{−1}; CFT, superposition model (SPM), angular overlap model (AOM)]. Electronic transitions between the d (or f) states localised at cations and split by the crystal field due to surrounding ligands.
Colour centres [25 000–4000 cm^{−1}; CFT]. Electronic transitions between the states localised at electron or hole point defects giving rise to absorption in the visible spectral range.
Fundamental vibrations, overtones and combination modes of molecules or charged atomic groups [8000–400 cm^{−1}; vibrational theory of isolated groups]. Refer, e.g., to the respective chapters in this volume.
Band–gap transitions [25 000–4000 cm^{−1}; band theory]. Transitions between bands in conductors, semiconductors, and doped semiconductors.
Most of the processes mentioned above occur in an energy range that covers the visible part of the electromagnetic spectrum, and are therefore responsible for the colour of many minerals and synthetic compounds. In particular, crystal field d–d transitions – which will be discussed in details in Chapter 3 – play a decisive role. In addition, MMCT processes are of great importance in mineralogy, since practically all common rockforming ferromagnesian mineral groups (as well as several other minerals) are affected to some extent. There, homonuclear Fe^{2+}–Fe^{3+} CT is the prevailing mechanism and often dominates the visible part of the spectrum with broad and intense absorption bands below 20 000 cm^{−1}. Furthermore, heteronuclear Fe^{2+}–Ti^{4+} CT has also been identified in many minerals. Besides, MMCT absorptions due to several other combinations of cations amenable to lowenergy redox processes have been discussed for minerals and are often observed, e.g., in metalorganic clusters and complexes. Since the present review focusses on crystal field aspects, we will not consider MMCT in details here but refer the reader to relevant basic literature (e.g. Smith & Strens, 1976; Smith, 1977; Burns, 1993) and references therein.
Beside the processes listed above, other physical mechanisms also determine colour: e.g. dispersion, scattering, interference and diffraction. A general overview to processes causing colour in minerals is given, e.g., by Nassau (1978, 1983).
Optical (UVVIS) spectroscopy of transition metal ion bearing compounds is a powerful experimental method used to investigate specific properties of both technologically useful crystals for industrial applications (e.g. laser materials etc.) as well as relevant minerals in geosciences (see below). A complementary technique, namely, electron magnetic resonance (EMR) spectroscopy, requires a separate consideration and will not be dealt with here. The readers may refer to the recent review articles (Rudowicz, 1987; Rudowicz & Misra, 2001; Rudowicz & Sung, 2001; Rudowicz, 2002) and books cited therein. For the understanding of the experimental results obtained by the optical as well as EMR spectroscopy methods, knowledge of the orbital energy levels and state vectors of the respective d or f transition ion is required. Crystal or ligand field theory (in the following: crystal field theory, CFT) provides a very successful formalism to quantify these energy levels and state vectors in terms of various sets of the freeion and CF parameters. As universally expressed by Newman & Ng (1989), these “crystal field (CF) parameters measure the strength of the interaction between the openshell electrons of paramagnetic ions and their surrounding crystalline environment”.
CFT is a model of chemical bonding applicable to transition metal and lanthanide ions, i.e., elements where d or f orbitals are subsequently filled with electrons (e.g., Schläfer & Gliemann, 1967; Lever, 1984). CFT describes the origin and consequences of interactions between the d or f orbitals with the electrostatic fields originating from negatively charged anions or dipolar groups coordinating the central ion. The CFT approach was pioneered by Bethe (1929) and van Vleck (1932) and provided the basis for the first successful analysis of the electronic spectra of transition metal ion bearing complexes.
In the early fifties, the importance of crystal field considerations became evident in view of the inconsistencies in interpretation of several experimental results. For example, it was shown that the heat of hydration of 3d^{N} ions as well as the lattice energies of 3d^{N}metal sulfides could only be explained when the crystal field stabilisation energy (CFSE) of the respective 3d^{N} ion was taken into account. These results stimulated many geoscientists to suggest that the CFSE should also affect or even account for the thermodynamic properties of 3d^{N}ion bearing minerals. Thus the importance of the CFSE for explanation of the intra and intercrystalline distribution behaviour of 3d^{N} ions as well as their partitioning between crystals and melts was recognised.
Applications of crystal field theory in geosciences
Widespread applications of CFT over the years have led to successful interpretation of a wide range of physical and chemical properties of transition metal bearing minerals. It was shown that the partially filled d orbitals of these ions contribute to specific geochemical, crystal chemical, structural, thermodynamic, magnetic, and spectral properties. Comprehensive reviews, e.g., by Burns (1970, 1993) and Langer (1988, 1990) illustrate the great advances in understanding of these properties since the 1950's, when CFT was first applied to problems in geosciences. Due to applications of CFT, several important problems could be solved as outlined briefly below.
a) Concepts of colour and pleochroism
The most widely recognised influence of the transition metal ions on the properties of minerals concerns their colour as evidenced by a number of reviews about the origin of colour (e.g., Platonov, 1976; Nassau, 1978; Marfunin, 1979; Rossman, 1988). In mineralogy and geochemistry, CFT in conjunction with absorption spectroscopy measurements is used to understand colour and pleochroism due to the d–d transitions in certain minerals.
b) Interpretation of electronic structure and bonding
Detailed interpretation and assignment of polarised electronic absorption spectra in the context of CFT enables the extraction of various CF and interelectronic repulsion parameters, giving insight into the electronic structure and bonding character prevailing in transition metal complexes in minerals (Lever, 1984). Such particularly valuable information can be extracted in systematic studies on solid solution series or synthetic host matrices doped with predefined concentrations of transition metal ions.
c) Partitioning of transition metal ions
i) Intercrystalline distribution. CFT was quickly applied to explain the partitioning of transition metals between solid silicates and silicate melts. A central concept in these interpretations is the CFSE. Reasonable assumptions about the nature of coordination sites in melts together with crystallographic knowledge of the site symmetry in silicate minerals yielded CFSE differences between the melt and solid constituents, which could be correlated with the observed element fractionations in slowly cooled magma chambers (Williams, 1959; Burns & Fyfe, 1964; Matsui et al., 1977).
Understanding of the partitioning of elements between closely related sites in solids requires more detailed information on the geometry of the crystallographic sites and the energy splittings of the metal d orbitals (Burns & Fyfe, 1966, 1967a, 1967b). Energy splittings were acquired from electronic absorption spectra for a variety of silicate minerals and were shown to yield values of CFSE in qualitative agreement with the known metal site partitioning (Schwarcz, 1967; Burns, 1993). Refinement of this model has proved to be difficult. So far, only the distribution behaviour of Cr^{3+}, occupying relatively undistorted octahedral positions in the respective structures, has been quantitatively correlated (Langer & Andrut, 1996).
ii) Intracrystalline distribution. Site occupancies of transition metal ions in minerals have been detected from intensities of the absorption bands in their polarised spectra (see e.g. Goldman & Rossman, 1979; Hu et al., 1990). It has been shown that this method is a powerful tool in addition to Mössbauer spectroscopy and Xray structure determination. A better insight into the dynamic behaviour was also achieved using elevated temperatures and pressures. Further applications of in situ measurements for determination of the kinetics of cation ordering have been reported (Burns, 1993).
d) Evaluation of thermodynamic properties of minerals influenced by CFSE
This concerns for instance the CFSE contributions to the electronic entropy or enthalpy of mixing.
e) Mantle geochemistry of transition elements
This concerns for instance polyhedral bulk moduli from highpressure spectra (Langer et al., 1997), phase transitions (Burns & Sung, 1978) and geophysical properties of the Earth's interior (radiative heat transport, electrical conductivity).
f) Detection of structural details
This concerns for instance cation ordering, phase transitions (Percival & Salje, 1989; Percival, 1990), dynamic or static JahnTeller distortions, polyhedral distortions (Goldman & Rossman, 1976). Recent investigation on the local structure of CrO_{6} polyhedra in solid solutions are presented in Part II.
Optical spectra
Most commonly observed lightrelated phenomena, like reflection, refraction, interference, diffraction and polarisation, can be explained by the nature of transverse electromagnetic waves. The evidence for the description of light as waves was well established at the turn of the 20^{th} century. Then the photoelectric effect provided also a firm evidence of the particle nature of light. Thus, light is said to exhibit a waveparticle dual nature. In different types of phenomena observed with light, either the wave picture or the particle picture is more appropriate for description.
Since the electromagnetic spectrum covers a few orders of magnitudes of energy, it is conveniently subdivided according to the way in which the various types of radiation are generated and used (see e.g. the introductory chapter in this book by Geiger, 2004). Thus, different units are commonly used in the field of spectroscopy, because the standardisation to one particular unit is not convenient for the description of all existing quantities.
The results of a spectroscopic experiment are generally represented in form of a spectrum, which visualises the intensity and energy of optical transitions induced by a certain process as a function of the electromagnetic wave frequency (refer to the spectral examples given in this chapter). Each absorption band corresponds to the transition of an electron from the ground state to one of the (possible) excited states. The position of the band in the spectrum complies with the energy of the respective transition due to a given process, while its integral intensity is correlated with the transition probability. The respective appropriate units used for description of optical spectra are summarised below.
Units used in optical spectroscopy
Measurement of the optical transition energies
In the literature the electromagnetic spectrum is either expressed in terms of energy E, wavelength λ or frequency v. For example, the visible part of the electromagnetic spectrum is generally described in terms of wavelength, commonly expressed in the SI units of nanometres [nm] according to the equation:
where c is the speed of light (in vacuum c_{0} = 2.99792458·10^{8} ms^{−1}) and v is the frequency expressed in number of oscillations per second [s^{−1}], that is in units of hertz [Hz]. However, in the field of optical spectroscopy the concept of the wavenumber = 1/λ, which is the reciprocal of the wavelength, is more widely adopted. The convenient unit is chosen as the reciprocal centimetre [cm^{−1}]. The reason for this choice stems from taking into account the wavelengthenergy relation. The photon energy E in joule [J] can be calculated according to Planck's formula as: with Planck's constant h = 6.626069·10^{−34} Js. Using Equation 1 we obtain:According to Equation 3 the wavenumber is directly proportional to the photon energy E, and therefore it is an appropriate quantity at the abscissa for the representation of an optical spectrum. The wavenumber unit of reciprocal centimetre [cm^{−1}] has proved to be convenient and easy to handle, since the values encountered in various optical spectroscopy techniques range, in general, between 40 000 and 4000 cm^{−1}. In the physics literature the photon energy E is sometimes also expressed in units of electron volt [eV] or atomic (energy) units [au]: 1 eV = 1.602176·10^{−19} J; 1 au = 4.359744·10^{−18} J (all constants taken from Mohr & Taylor, 2000). Table 1 provides some numerical relationships between wavenumber, wavelength, frequency and energy in the range relevant for optical spectroscopy. In case a given quantity is related to 1 mole, it has to be multiplied by Avogadro's number of particles per mole, N_{A} = 6.022142·10^{23} mol^{−1}. Note that Taylor (1995) gives an overview of the use of the international system (SI) of units.
wavenumber [cm^{−1}]  wavelength λ [nm]  frequency v [10^{14} Hz]  energy E [10^{19} J]  energy E [eV] 

40000  250  11.99  7.95  4.96 
25000  400  7.49  4.97  3.10 
18182  550  5.45  3.61  2.25 
14286  700  4.28  2.84  1.77 
10000  1000  3.00  1.99  1.24 
4000  2500  1.20  0.79  0.50 
wavenumber [cm^{−1}]  wavelength λ [nm]  frequency v [10^{14} Hz]  energy E [10^{19} J]  energy E [eV] 

40000  250  11.99  7.95  4.96 
25000  400  7.49  4.97  3.10 
18182  550  5.45  3.61  2.25 
14286  700  4.28  2.84  1.77 
10000  1000  3.00  1.99  1.24 
4000  2500  1.20  0.79  0.50 
Measurement of the intensities of the transitions
Consider a monochromatic beam of intensity I_{0} with a perpendicular incidence onto a sample with planar parallel faces and a thickness d. Absorption takes place in the sample and the beam of radiation leaves the sample with a smaller intensity I, while a fraction of light with the intensity I_{a} is absorbed by the sample. The following definitions assume that the entire incident light is either transmitted or absorbed, while reflection or scattering is negligible. The amount of radiation absorbed may be then described by several quantities defined below:
Note that the logarithm to the base of 10 applied here is commonly used in chemistry and geosciences, while physicists usually adopt the natural logarithm. The following experimental laws describe the absorption or transmission properties of samples.
BouguerLambert law.
According to the BouguerLambert law the absorbance A is proportional to the absorption path length d (i.e., the sample thickness):
where α is the linear (decadic) absorption coefficient in [cm^{−1}]. The quantity α depends on the energy of radiation and is specific for a given material. The BouguerLambert law, expressed by Equation 8, when inserted into Equation 7 yields:BeerLambertBouguer law.
The BouguerLambert law in Equation 8 may be expanded to describe also the dependence on the concentration c of the absorbing species, i.e., expressed in moles per volume, it yields:
The proportionality constant ε is now called the (linear) molar (decadic) absorption coefficient. It is also specific for a given material and depends on the energy of radiation. Using the thickness d in [cm] and concentration c either in [mol l^{−1}] or [mol cm^{−3}], yields ε in the units of [l mol^{−1} cm^{−1}] or [cm^{2} mol^{−1}]. By inserting Equation 10 into Equation 7, we obtainInstrumentation
A spectroscopic experiment requires a light source, some means of providing the energy resolution of the radiation before or after interaction with the sample, and a detection system. The use of appropriate polarisers allows studying the directiondependent absorption behaviour of anisotropic solids.
The instrumentation used in optical spectroscopy can be classified according to the means of the energy resolution applied into dispersive and Fouriertransform spectrometers (Burns, 1970; Langer & Frentrup 1979; Griffiths & de Haseth, 1986; McMillan & Hofmeister, 1988). In the last decades, there has been a boost in the development and application of Fouriertransform infrared (FTIR) spectrometers. More recently, the accessible spectral range for commercial FT spectrometers was expanded from the infrared up to the nearUV range due to introduction of new interferometers, light sources, and detectors. A high local resolution down to the order of 20 μm, especially useful in the field of geosciences, was achieved by attaching a microfocusing unit or a mirroroptics microscope to the spectrometer.
As a consequence, even microscopebased dispersive instruments with comparable local resolution lost some popularity due to their inherent disadvantages (e.g., no internal frequency calibration, lower spectral resolution, lower signaltonoise ratio; cf. Griffiths & de Haseth, 1986). For certain applications dispersive instruments are still favoured (e.g. Raman and emission spectroscopy; see the respective chapters by Nasdala et al., 2004a, 2004b in this book).
In addition, microscope spectrometric techniques allow the usage of heating/cooling stages to cover a wide temperature range as well as of the diamond anvil cells (DAC) to provide high pressure (several tens of GPa; Ferraro, 1984; Langer, 1990). The current developments of DAC's concentrate on attaining high pressures and high temperatures simultaneously in order to enable studies of the processes occurring deep in the Earth's interior. The hydrothermal DAC is designed to obtain direct structural and thermodynamic information on geochemical fluids (Bassett et al., 1993).
Principles of crystal field theory
Recommended textbooks
A selection of textbooks for beginners and advanced readers covering CFT as well as other relevant aspects of spectroscopy comprise, e.g., Bersuker (1996), Burns (1970, 1993), Cotton (1990), Figgis & Hitchman (2000), Gade (1998), Gerloch & Constable (1994), Henderson & Imbusch (1989), Henderson & Bartram (2000), Hüfner (1978), Lever (1968), Lever (1984), Marfunin (1979), Morrison (1992), Mulak & Gajek (2000), Newman & Ng (2000a), Powell (1998), Schläfer & Gliemann (1967), or Solomon & Lever (1999). Further textbooks covering specific aspects are cited within the text.
The description of the interaction of the d (or f) electrons of a transition ion with the crystal field generated by the surrounding ligands requires background knowledge of quantum mechanics (QM). Basic ideas indispensable for understanding the crystal field theory are succinctly presented in this section. We start with oneelectron systems and proceed to manyelectron systems. Then we deal with the selection rules, which require some knowledge of group theory, and the more advanced experimental topic of the d–d absorption bands. Finally the qualitative appraisal of the major parameters involved in CFT is discussed.
Oneelectron systems
Schrödinger equation and quantum numbers
The localised d or f electron can be treated as a stationary system, i.e., independent of time. Hence, its energy is described by the timeindependent oneelectron Schrödinger equation:
where H is the socalled Hamiltonian, i.e., the operator of the total energy of the system, and E is the total energy corresponding to the state (Ψ) under consideration. In quantum mechanical parlance Equation 12 is an eigenvalue equation, i.e., the operator H acting on the wavefunction ψ yields a set of numbers (i.e., E) multiplied by the same wavefunction, i.e., an eigenfunction. Here the numbers, i.e., the eigenvalues of H, represent the energy corresponding to a given wavefunction. Each wavefunction describes a certain state of the system, which is responsible for the observed properties of the system. Notably the Ψ^{2}=∫Ψ^{*};Ψdr describes the spatial probability of finding the electron in the given volume element over which the integration is taken. In practice Ψ is normalised to unity, i.e., integrating over all space should yield Ψ^{2}≡1. Having included all physical interactions relevant for a given system in its Hamiltonian H, the major problem is how to solve Equation 12, i.e., how to find Ψ (wavefunction) and the corresponding E (energy). It is not an easy matter and the reader may consult quantum mechanics textbooks for some simple worked examples. Based on this knowledge we here proceed to describe our case study system, i.e., a d electron in a transition metal ion with the d^{1} configuration. The motion of a d electron in a free ion, referred to a spherical coordinate system (r, θ, φ), can be described by The first term (in square brackets) in the Hamiltonian in Equation 13 represents the kinetic energy of the electron, where ∇^{2} is the Laplacian operator (e.g. in one dimension ), ħ is Planck's number (where ħ = h/2π), and m is the electron mass. The second term represents the potential energy of the d electron due to the nucleus and all other electrons (e.g. in the closed shells) in the ion. As a first approximation the potential V(r) can be assumed to be spherically symmetric: V(r) = V(r), i.e., depending only on the distance of the d electron from the nucleus and not on the direction in space. This equation can be solved exactly for one d electron and historically it was first done for the simplest oneelectron system, i.e., the hydrogen atom. Using the method of separation of the spatial variables, it can be shown that the solutions of Equation 13 are the singleelectron hydrogenlike wavefunctions: Such functions comprise the radial function R(r), the spherical harmonics Y(θ, φ), and the spin function χ, which is independent from the spatial coordinates of the electron. Note that the spherical harmonics Y_{lml}(θ, φ) are the eigenfunctions of the angular momentum operators l^{2} and l_{z} – for details see QM books. The various solutions describing the possible states of oneelectron systems, starting from the ground state (n = 1, l = m_{l} = 0, m_{s} = ±1/2) and all excited states lying higher in energy are parameterised by the four quantum numbers (n, l, m_{l}, m_{s}). The meaning of each quantum number is briefly described below (for details and derivations the reader is referred to standard quantum chemistry textbooks).1) The principal quantum number n is an “index” for the radial function R_{n}(r), which describes the average distance of the electron from the nucleus. It is directly related to the overall energy of an electron in a given n^{th} orbit: E ∝ –1/n^{2}. The number n is an integer ranging from 1 to ∞ and corresponds to the electron orbits or shells: K (n = 1), L (n = 2), M (n = 3), N (n = 4) etc.
2) The orbital quantum number l describes the orbital momentum of the electron. It reflects the shape of the corresponding orbital with the energy increasing as E ∝ l. The solutions of the angular momentum equations require that l can have integer values between 0 and n – 1 corresponding to the orbital types: s (l = 0), p (l = 1), d (l = 2), f (l = 3) etc. Consequently, the K shell can host only the s orbital, the L shell the s and p orbitals, the M shell the s, p, and d orbitals, and so on. Each orbital type has a specific spatial symmetry, which describes the electronic density distribution in an ion, e.g., the s orbitals are spherical, p orbitals dumbbellshaped, and d orbitals doubledumbbell shaped.
3) The magnetic quantum number m_{l}, which describes the orientation of the orbitals with respect to a reference direction, can have an integer value between –l, –l + 1, …, 0, …, l – 1, and l. The number m_{l} thus determines the number of orbitals that can be occupied by electrons in a shell with a given value of n: one s orbital (in all shells, i.e., n ≥ 1), three p orbitals (n ≥ 2), five d orbitals (n ≥ 3), seven f orbitals (n ≥ 4) etc. In the absence of an electromagnetic field, the orbitals with the same n and l but different m_{l} are equal in energy and are said to be “degenerate”, i.e., several wavefunctions corresponds to one energy level. The shape and orientation of the s, p, and d orbitals are depicted in Figure 1, together with their group theoretical representations for cubic O_{h} symmetry (see below). With regard to the CF concept it is important to note that the d_{xy}, d_{yz} and d_{xz} orbital lobes are located between the respective Cartesian reference axes, whilst d_{x2}_{–y2} and d_{z2} are directed along these axes. This determines how the energy of the orbitals change if a d^{1} ion is placed in a crystal field (see later).
4) The spin quantum number m_{s} can take the values +½ and −½ corresponding to the parallel or antiparallel orientation of the electron spin angular momentum. The spin s is a purely quantum mechanical quantity and unlike the orbital angular momentum l has no classical equivalent. Each orbital described by the quantum numbers (n, l, m_{l}) can host two electrons with m_{s} = ± ½. Hence, the maximum occupancy of a given l orbital is at most two s electrons, six p electrons, ten d electrons, 14 f electrons etc. per shell.
Aufbau principle, Pauli exclusion principle and Hund's rule
The quantum numbers (n, l, m_{l}, m_{s}) have provided the quantum mechanical basis for the periodic table of elements originally based on the experimental observations, e.g. Hund's rules. One more concept is needed to understand how the orbitals are filled with electrons according to the Aufbau principle in the order of increasing energy. Electrons must obey the Pauli exclusion principle, which states that within one atom no electrons with the same set of four quantum numbers may occur. According to Hund's rules, the degenerate l orbitals are filled preferably up to halfoccupancy with the electrons with parallel spin; only subsequently electrons are paired with antiparallel spin, thus leading to maximum spin multiplicity of the ground state. Closed, i.e., completely filled and hence spherical shells like the argon core [Ar] = 1 s^{2}2s^{2}2p^{6}3s^{2}3p^{6} in transition ions contribute only to the spherical part of the potential V(r) in Equation 13 acting on the nd^{N}^{} electrons. In case of the transition elements, the openshell d or f orbitals are successively filled with electrons. Table 2 summarises the electronic configurations for the metals of the 3d series and their common cations. Ionisation is accomplished by removal of the 4s electrons and afterwards some 3d electrons.
Z  element  M^{0}  M^{+}  M^{2+}  M^{3+}  M^{4+}  M^{5+}  M^{6+}  M^{7+} 

21  Sc  [Ar] 3d^{1}4s^{2}  [Ar]  
22  Ti  [Ar] 3d^{2}4s^{2}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
23  V  [Ar] 3d^{3}4s^{2}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
24  Cr  [Ar] 3d^{5}4s^{1}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar]  
25  Mn  [Ar] 3d^{5}4s^{2}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{1}  [Ar]  
26  Fe  [Ar] 3d^{6}4s^{2}  [Ar] 3d^{6}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{2}  
27  Co  [Ar] 3d^{7}4s^{2}  [Ar] 3d^{7}  [Ar] 3d^{6}  
28  Ni  [Ar] 3d^{8}4s^{2}  [Ar] 3d^{9}  [Ar] 3d^{8}  [Ar] 3d^{7}  [Ar] 3d^{6}  
29  Cu  [Ar] 3d^{10}4s^{1}  [Ar] 3d^{10}  [Ar] 3d^{9}  [Ar] 3d^{8}  
30  Zn  [Ar] 3d^{10}4s^{2}  [Ar] 3d^{10} 
Z  element  M^{0}  M^{+}  M^{2+}  M^{3+}  M^{4+}  M^{5+}  M^{6+}  M^{7+} 

21  Sc  [Ar] 3d^{1}4s^{2}  [Ar]  
22  Ti  [Ar] 3d^{2}4s^{2}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
23  V  [Ar] 3d^{3}4s^{2}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
24  Cr  [Ar] 3d^{5}4s^{1}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar]  
25  Mn  [Ar] 3d^{5}4s^{2}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{1}  [Ar]  
26  Fe  [Ar] 3d^{6}4s^{2}  [Ar] 3d^{6}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{2}  
27  Co  [Ar] 3d^{7}4s^{2}  [Ar] 3d^{7}  [Ar] 3d^{6}  
28  Ni  [Ar] 3d^{8}4s^{2}  [Ar] 3d^{9}  [Ar] 3d^{8}  [Ar] 3d^{7}  [Ar] 3d^{6}  
29  Cu  [Ar] 3d^{10}4s^{1}  [Ar] 3d^{10}  [Ar] 3d^{9}  [Ar] 3d^{8}  
30  Zn  [Ar] 3d^{10}4s^{2}  [Ar] 3d^{10} 
Crystal field splitting – an overview
If a free transition metal ion is inserted into a hypothetical field of negative charges evenly distributed on a sphere, the five d orbitals will be raised in energy (i.e., destabilised) due to the interaction of the d electrons with the negative field, but remain degenerate. However, in the case when the negative charges are more or less localised at certain ligand atoms in a crystal, glass or solution, the resulting (nonspherical) crystal field (CF) potential then splits the five d orbitals into the orbital groups or single orbitals of different energy. These energy splittings are described by the CF parameters Δ_{i} or 10Dq; in particular, in cubic symmetry CF Δ_{i} = 10Dq, the subscript i refers to the polyhedral type (see below). The magnitude and type of such crystal (or ligand) field splittings within the five d orbitals depend on the symmetry and geometry of the coordination polyhedron (i.e., on the polar coordinates of the ligand atoms) as well as on the actual type of the central transition ion and the ligand atoms. For a given central ion (M) and ligand (L) polyhedron ML_{x}, Δ_{i} (10Dq) strongly increases with decreasing M–L bond lengths R, theoretically by 1/R^{5} (e.g. Schläfer & Gliemann, 1967). The origin of this dependence will be discussed in Part II.
In the following, we consider examples of several coordination polyhedra, which are most often found in crystal structures. For regular polyhedra with high local site symmetry even simple geometrical considerations reveal the correct type of the CF splittings. For example, in a regular octahedral coordination (symmetry O_{h}) six ligands are located at the same distance on the axes of the local Cartesian coordinate system. As compared to a spherical field, those orbitals with their lobes pointing in between the Cartesian axes, i.e., d_{xy}, d_{yz}, and d_{xz}, will be lowered in energy and hence stabilised, while the d_{x}^{2}–_{y}^{2} and d_{z2} orbitals pointing along the axes directly towards the ligands will be raised in energy and hence destabilised. The former orbital group is triply degenerate (denoted by “t” for triplet) and the signs of their wavefunctions (Fig. 1) are symmetrical with respect to the octahedral centre of inversion (denoted by “g” for German “gerade” = even), resulting in the group theoretical representation t_{2g} (subscript 2 can be ignored for the moment). The latter orbital group is double degenerate (denoted by “e” for doublet) and also obeys the octahedral inversion symmetry (g), leading to the representation e_{g}. Here the lower case symbols for orbitals and representations refer to the oneelectron quantities, while the upper case symbols will be used to denote manyelectron quantities discussed in the next chapter. Since a centre of gravity rule has to be applied, it follows that the total octahedral (subscript o) CF splitting Δ_{o} comprises a stabilisation of the three t_{2g} orbitals by 2/5 Δ_{o} (4Dq) and a destabilisation of the two e_{g} orbitals by 3/5 Δ_{o} (6Dq). The octahedral situation is depicted on the righthand side of Figure 2. These quantitative results are obtained by solving the eigenequation, like Equation 12, for the cubic CF Hamiltonian (see textbooks).
In a tetrahedral (regular symmetry T_{d}) or cube coordination (O_{h}) the d_{xy}, d_{yz}, and d_{xz} orbitals point towards the edges of the cube (distance to ligand a/2), the d_{x}2_{–y}2 and d_{z}2 orbitals towards the cube faces (at the distance a). Hence the CF splitting is inverted as compared to the octahedral case, obeying the centre of gravity rule. Since the tetrahedron lacks a centre of symmetry, the signs of the wavefunctions with respect to a centre of symmetry are formally ignored, and therefore the g indices are omitted from the representations (t_{2}, e). Important implications of this fact will be discussed later. Furthermore, it can be shown that the tetrahedral (subscript t) splitting Δ_{t} is only a fraction of the octahedral one: Δ_{t} = – 4/9 Δ_{o} (given the same ions and M–L distances). In case of the centrosymmetric cube (subscript c) coordination, the coordination number and hence the CF strength is doubled, i.e., Δ_{c} = 2 Δ_{t} = – 8/9 Δ_{o}. The CF splittings for tetrahedra and cubes are shown on the lefthand side of Figure 2.
Such considerations can also be done for other, less common regular coordinations. Upon reduction from cubic symmetry, the degenerate orbitals (t and e) split further into either twofold degenerate (e) or nondegenerate levels denoted by the letter a (symmetric with respect to the principal axis) or b (antisymmetric). Various examples of orbital splittings in regular coordinations are shown in Figure 3 in units of Dq. The respective calculations are based on constant M–L distances and identical ligands. Since the polyhedra in Figure 3 have high axial symmetries, the orbital doublet (d_{xz}, d_{yz}) remains degenerate in all cases considered. Note that in some cases angular variations (under maintained symmetry) may change the relative orbital energies.
In the case of a distortion of regular polyhedra (and further lowering of symmetry), the resulting sequence of orbitals can then be estimated only for highsymmetry distortions, e.g., for octahedra elongated or compressed along a four or threefold axis. In order to elucidate the respective sequences for irregular distortions, as well as to obtain quantitative results, advanced calculation methods have to be applied, some of which are outlined in Part II. Evidently, for polyhedra distorted from cubic symmetry a single CF parameter like Δ_{o} or Δ_{t} is not sufficient to describe adequately the resulting splittings of the d orbitals. The forms of CF Hamiltonian and the nonzero CF parameters appropriate for various symmetry cases will be discussed later.
At this stage of discussion, the reader should keep in mind the definition cited in section 1.1 that CF parameters represent a measure of the d (or f) electron interaction with the (whole) surrounding crystalline environment. Hence, this interaction is generally not solely restricted to the first polyhedral coordination sphere. On the one hand, such nextnearest neighbour contributions can quantitatively modify the CF parameters to a certain extent. On the other hand, a reduced symmetry of the longrange crystalline environment will also qualitatively affect the orbital splittings. Hence, in the discussion of level splitting schemes, the actual crystallographic site symmetry of a transition metal ion overrides an incidental higher pseudosymmetry of the local polyhedron. However, as outlined later in section 3.4.4., the consideration of local pseudosymmetries is often indispensable for a meaningful interpretation of optical absorption spectra.
As a direct consequence of the d orbital splittings and the resulting, at least partial, removal of degeneracy due to the CF, these orbitals are no longer filled with electrons in an arbitrary way as in a free ion, but in order of increasing energy. We note that for 3d^{N} ions in minerals and similar systems with oxygen ligands at ambient conditions, Hund's rule remains valid upon successive filling of the d orbitals with electrons. This yields the observed highspin (HS) configuration with the maximum possible spin multiplicity. The HS cases are realised if the spin pairing energy P (i.e., the energy needed to combine two electrons with antiparallel spin into one orbital) is larger than the CF splitting Δ. In the strong crystal fields Δ often exceeds P, and then the d electrons are primarily paired in the lowlying d orbitals (i.e., Hund's rule is violated). Such lowspin (LS) configurations yield the reduced magnetic moments and spin multiplicity. The LS cases are relevant for certain d^{N} ions only, depending on the particular type and symmetry of the coordination polyhedron. Table 3 summarises the HS and LS configurations for octahedral and tetrahedral crystal fields. The LS configurations occur mainly for the heavier d elements (4d, 5d series). Among 3d^{N} cations, octahedral Co^{3+} tends to adopt the lowspin state even with oxygen ligands. In coordination of ligands producing strong crystal fields like CN^{−}, or under high pressures, Co^{3+} as well as Fe^{2+} are expected to occur in the LS configuration. The electronic, magnetic, and structural properties of various cations in the HS and LS configuration differ significantly. Hence, the pressureinduced HS to LS phase transitions of Fe minerals may play an important role in explaining the physical peculiarities in the Earth's mantle (see e.g., Cohen et al., 1997; Burns, 1993, and references therein). For systems close to the HS–LS crossover point, the thermal and lightinduced spin transitions were also investigated (see e.g. Gütlich et al., 1999).
JahnTeller effect
Simple geometrical consideration, like those above which elucidated the CF splittings for regular coordinations shown in Figure 2, cannot explain specific complications arising for particular d^{N} electron distributions, especially evident in octahedral coordination. According to the JahnTeller theorem, a molecular system with a degenerate electronic ground state will distort to lower its symmetry and reduce the energy, thereby removing the degeneracy. Recalling the fact that the degenerate e_{g} orbitals d_{x2–y}2 and d_{z2} are directed towards the octahedral ligands, it follows that the uneven electron distributions among these orbitals in the highspin d^{4}, d^{9} and lowspin d^{7} configurations lead to specific electronligand interactions. This promotes a lifting of degeneracy for the e_{g} orbitals, accomplished by a generally strong static distortion (and symmetry reduction) of the octahedral coordination. These distortions are well established experimentally for Cr^{2+}(d^{4}), Mn^{3+}(d^{4}) and especially Cu^{2+}(d^{9}) by numberless crystal structure investigations (for reviews see e.g. Shannon et al., 1975; Effenberger, 1986; Norrestam, 1994) as well as spectroscopic studies. Note that CFT cannot predict the sense of the octahedral distortion (i.e., elongation or compression). However, most d^{4} and d^{9} ions yield strongly elongated coordinations (i.e., dipyramidal 6fold, onesided pyramidal 5fold or planar 4fold ones), whereas compressed dipyramidal coordinations are found only occasionally. Less pronounced distortions are expected in tetrahedral environments for d^{N} ions with uneven t_{2} electron distribution. However, examples where tetrahedral distortion can be definitely attributed to the CF effects are scarce and mostly restricted to Cu^{2+} (3d^{9}). In principle, JahnTeller distortions and even their type can also be postulated for uneven electron distribution in the octahedrally stabilised t_{2g} orbitals. Such effects are usually too small to induce static distortions but manifest themselves by dynamic effects, generally a promotion of JahnTelleractive polyhedral vibrational modes (e.g., Bersuker, 1984). For example, statistical evaluation of a large number of distorted Co^{2+}O_{6} octahedra (3d^{7}) gave no significant hint for a preferred distortion to elongated dipyramids as predicted by the CF considerations (Wildner, 1992).
Crystal field stabilisation energy (CFSE)
Another important consequence of the specific d electron distributions arising from CF considerations as listed in Table 3 is an energetic stabilisation of most d^{N} transition ions in a CF as compared to the average energy of all five d orbitals. In a regular octahedral CF, for example (cf. Fig. 2), each electron incorporated into the energetically lower t_{2g} orbitals stabilises the transition metal complex by 2/5 Δ_{o}; on the other hand, electrons in the e_{g} orbitals destabilise the complex by 3/5 Δ_{o} each. The resulting net energy gain is called the crystal field stabilisation energy (CFSE) and is listed in Table 3 for octahedral and tetrahedral coordination. Table 3 shows that the empty d orbitals (3d^{0}), halffilled d orbitals (3d^{5}, HS), and completely filled d orbitals (d^{10}) are not further stabilised (CFSE = 0), whereas all other possible configurations gain energy in a CF. Consequently, the CFSE has an important impact on the properties of transition metal compounds, e.g., thermodynamic properties (e.g. Burns, 1985, 1993; Langer, 1988).
A particular aspect of the CFSE is related to the intracrystalline distribution of transition metal ions among available sites in a crystal structure, glass phase or melt, as well as to the intercrystalline distribution among various coexisting phases during the crystallisation process, or during fractionation between crystal and melt. Despite the fact that the CFSE constitutes only a minor contribution in regard to the total energy of a system, it can be the decisive force for the preferred incorporation into the structural site A vs. site B, or into phase C vs. phase D.
Intra and intercrystalline transition ion distribution
A wellestablished example of the crucial influence of the CFSE on the intracrystalline distribution of transition ions in octahedral and tetrahedral sites is the spinel group compounds (see e.g.Burns, 1993, and references therein). For a particular XY pair, the formation of a “normal” or “inverse” spinel type is often predicted correctly from the differences between the respective octahedral and tetrahedral CFSE. These differences are also expressed as “octahedral site preference energy”, OSPE (see e.g.Burns, 1993). A much more common case in many mineral structure types is the presence of several crystallographically distinct sixfold coordinated sites, which may differ in symmetry, ligand types, size (average M–L bond lengths), and distortion (individual bond lengths and angles). Although a prediction of the transition ion site distribution solely by the crystal field arguments is often rather ambiguous, nevertheless, such considerations can be extremely helpful in the interpretation of experimental results.
Concerning the intercrystalline transition metal distribution, different CFSE's for the 3d^{N} ions in phases with different cation–ligand distances and angles as well as different site symmetry may force the 3d^{N} ions predominantly into one of the phases in multiphase assemblages. CFT predicts that 3d^{N} ions preferentially fractionate into the phase with the highest CFSE for a suitable site (e.g. Burns, 1970, 1993, and references therein). To understand the influence of the CFSE on the distribution behaviour, quantitative relations between the transition metal concentration and the respective CFSE have to be investigated. In particular, any effects of the bulk chemistry, temperature and pressure changes on the partitioning of 3d^{N} ions in a given geochemical system, e.g. the minerals in the rock, have to be ruled out. As an example, Langer & Andrut (1996) investigated the distribution of Cr^{3+} ions (3d^{3}) among paragenetic coexisting minerals in various rocks from the upper mantle. The Cr^{3+} ions generally exhibit the highest CFSE's among the geochemically significant 3d^{N} ions. The results quantitatively confirmed the CF concept for a variety of parageneses by showing that the Cr^{3+} concentration is positively correlated with its CFSE in the respective mineral phase. For one particular paragenesis, however, a negative correlation was observed. In this case, the partition behaviour of Cr^{3+} ions was governed by a coupled substitution between the coexisting phases. The results show that the CF concept often succeeds in explaining the partition behaviour of transition metals between various coexisting phases, but several other competing mechanisms like crystal chemical effects also have to be taken into account.
Electronic transitions
Having introduced the basic concept of CF splittings as illustrated in Figure 2, we can now consider the electronic transitions from the ground d orbital level to the excited ones. Such transitions may be promoted by absorption of energy corresponding to Δ_{i} (i.e., Δ_{o}, Δ_{t}, or Δ_{c}) from the electromagnetic spectrum when the condition hv = Δ_{i} is satisfied. For certain reasons, which will be discussed later, such transitions usually give rise to more or less broad absorption bands (with approximated Gaussian shape) instead of narrow lines. The number of such transitions obeying Hund's rule (i.e., no spin change during transition) expected for the various d^{N} configurations may be derived from Table 3. For example, in the octahedral case d^{1}, d^{4}, d^{6} and d^{9} configurations should each give rise to one transition only (), d^{2}, d^{3}, d^{7} and d^{8} configurations to two transitions each and and and and , while for a d^{5} configuration no transition at all is predicted. For the tetrahedral field, the same number of transitions is expected, but for the reasons discussed above, with an inverted level sequence.
However, experimental electronic absorption spectra show (see e.g. spectra in Lever, 1984; Burns, 1993) that most predictions drawn from the hitherto simple CF considerations are not fulfilled: spectra of all systems from d^{2} to d^{8} often exhibit several additional (generally) weak absorption bands, and d^{2}, d^{3}, d^{7}, and d^{8} configurations all give rise to one further more intense band. Only spectra of the oneelectron d^{1} (Ti^{3+}) and “onehole” d^{9} (Cu^{2+}) systems comply with the expectation of a single absorption band (which is indeed mostly split due to the JahnTeller distortion of the ground (d^{9}) or excited (d^{1}) state). The single hole in the d^{9} configuration can be treated in analogy to the single electron in d^{1}, but yielding an inverted level sequence. The reason for the failure of this descriptive CF interpretation for d^{2} to d^{8} systems is that it relies solely on the oneelectron (i.e., hydrogenlike) orbital view and completely neglects interelectronic interactions in manyelectron systems (cf. e.g. Gerloch & Constable, 1994). Hence for a full understanding of the 3d^{N} ions in crystals, consideration of the latter interactions is indispensable.
Manyelectron concept
Manyelectron Hamiltonian
The approach outlined in the previous section served only an introductory purpose. We considered electrons in the d orbitals described by the oneelectron hydrogenlike wavefunctions. In order to provide sufficient basis for a thorough interpretation of optical absorption spectra of transition element compounds, we have to consider the manyelectron d^{N} systems taking into account the interelectron interactions (i.e. repulsions) between the d (and to a lesser extent f) electrons within an openshell orbital set. Even in the free transition ions, such electronelectron interactions lead to a splitting of the degenerate d orbital set into several spectroscopic terms. Hence, we have to introduce further perturbations into the manyelectron Schrödinger equation in addition to the CF. The full Hamiltonian H for the d^{N} electrons in crystals, instead of that in Equation 13, has the form:
In Equation 15 the first term, H_{spher}, denotes the freeion Hamiltonian in the spherical approximation, which is obtained using the HartreeFock central selfconsistent field (CSCF) approximation. The energy level obtained at this stage is the nd^{N} configuration, which is manyfold degenerate. The second term denotes to the electronelectron (ee) repulsion effects within the nd^{N} configuration not included in H_{spher}, i.e., the socalled residual electronic repulsion. This term yields the splitting of the nd^{N} configuration into the ^{2}^{S}^{+1}L multiplets. The third term comprises the crystal field interactions, which may be composed of several terms due to CF contributions of various symmetry. The last two terms represent the spinorbit (SO) and the electronic spinspin (SS) interactions, which lead to further splittings of the ^{2}^{S}^{+1}L multiplets or the CF energy levels. To illustrate the consecutive splitting of the energy levels, Figure 4 shows for a case study of 3d^{4} or 3d^{6} ions a schematic representation (not to scale) of the energy levels (EL) corresponding to different Hamiltonians in Equation 15. Basic notation for the wavefunctions (Ψ) is explained in the review by Rudowicz & Sung (2001). The distinction between the physical Hamiltonians and the effective (spin) Hamiltonian as well as the origin of the zerofield splitting (ZFS) as due to the perturbation V = H_{SO} + H_{SS} is also elucidated in Figure 4.Depending on the atomic number of the transition element, three cases of the relative importance of the CF term with respect to other interactions in terms of the respective energy (E), i.e. the overall energy level splittings, can be discerned:
Case 16a predominantly corresponds to the lowspin complexes of the 4d and 5d transition elements, where the CF splitting prevails over those due to other interactions; hence it is also referred to as the “strong field case”. In case 16b the splitting into spectroscopic terms is somewhat larger but of the same magnitude as the CF splittings (see also Fig. 4), whereas the splitting due to the spinorbit coupling can be ignored in a first approximation. This situation mainly applies to oxygenbased firstrow transition metal complexes (highspin) we are mostly interested in this review. With reference to the comparable strength (E_{ee} ≈ E_{CF}) it is also called the “intermediate field case”. In case 16c splittings due to the spinorbit coupling dominate; it is also called the “weak field case”. This situation applies mainly to the f transition element systems, giving rise to the particularly different appearance of their spectra and hence colours.
Below we will deduce the spectroscopic ^{2}^{S}^{+1}L terms (i.e., multiplets) for freeion 3d^{N} configurations and subsequently examine how these terms behave upon perturbation due to an intermediate CF (case 16b).
RussellSaunders coupling and freeion terms
Combining the oneelectron hydrogenlike orbitals into the wavefunctions suitable for the manyelectron system requires a method of the angular momentum theory. Coupling of the orbital and spin momenta of single electrons may follow either the LS (i.e. RussellSaunders) or jj coupling scheme. The LS coupling scheme is suitable for the lighter d elements, of interest to us, whereas the jj coupling scheme applies to the rare earth ions. Hence, the splitting of a 3d^{N} configuration into the freeion terms due to the residual electronic repulsion H_{ee} not included in H_{spher} may be described in the most convenient way (see also Fig. 4). In the LS coupling scheme the orbital angular momenta l_{i} (with the quantum numbers m_{l}) of all d electrons are coupled to yield the total orbital momentum L, as well as all spin angular momenta s_{i} (with the quantum numbers m_{s}) are coupled to yield the total spin S, i.e., we obtain the new quantum numbers and Analogously to the oneelectron orbital quantum number l, L is denoted by the capital letters S (L = 0), P (L = 1), D (L = 2), F (L = 3), G, H, I and so on. Spectroscopic terms are then labelled ^{2}^{S}^{+1}L_{J}, where J is the quantum number for the total angular momentum J and take all values differing by one between (L + S) and (L – S) (for our purposes, we omit J from now on). Depending on the spin multiplicity, which equals (2S + 1), the terms are called spin “singlet” (2S + 1 = 1), “doublet” (2S + 1 = 2), “triplet”, “quartet” and so on. It is important here to distinguish the degeneracy due to the orbital and spin variables. The total degeneracy of a ^{2}^{S}^{+1}L term is (2S + 1)(2L + 1), i.e., the product of its spin multiplicity (2S + 1) and its orbital multiplicity (2L + 1). The ground term for each d^{N} configuration can be deduced easily according to Hund's rule, since it must posses the maximum spin multiplicity. If there are more than one such term, the one with the highest total orbital momentum quantum number L lies lower in energy. Table 4 lists the respective quantum numbers and the resulting ground terms for d^{N} configurations.
m_{l}  

d^{N}  2  1  0  1  2  L = Σm_{l}  2L + 1  S = Σm_{s}  2S + 1  ground term (degeneracy) 
d^{1}  ↑  2  5  1/2  2  ^{2}D (10)  
d^{2}  ↑  ↑  3  7  1  3  ^{3}F (21)  
d^{3}  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28)  
d^{4}  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25)  
d^{5}  ↑  ↑  ↑  ↑  ↑  0  1  5/2  6  ^{6}S (6) 
d^{6}  ↓↑  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25) 
d^{7}  ↓↑  ↓↑  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28) 
d^{8}  ↓↑  ↓↑  ↓↑  ↑  ↑  3  7  1  3  ^{3}F (21) 
d^{9}  ↓↑  ↓↑  ↓↑  ↓↑  ↑  2  5  1/2  2  ^{2}D (10) 
m_{l}  

d^{N}  2  1  0  1  2  L = Σm_{l}  2L + 1  S = Σm_{s}  2S + 1  ground term (degeneracy) 
d^{1}  ↑  2  5  1/2  2  ^{2}D (10)  
d^{2}  ↑  ↑  3  7  1  3  ^{3}F (21)  
d^{3}  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28)  
d^{4}  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25)  
d^{5}  ↑  ↑  ↑  ↑  ↑  0  1  5/2  6  ^{6}S (6) 
d^{6}  ↓↑  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25) 
d^{7}  ↓↑  ↓↑  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28) 
d^{8}  ↓↑  ↓↑  ↓↑  ↑  ↑  3  7  1  3  ^{3}F (21) 
d^{9}  ↓↑  ↓↑  ↓↑  ↓↑  ↑  2  5  1/2  2  ^{2}D (10) 
In principle, it is not difficult to derive the excited freeion terms corresponding to various other possible arrangements of electrons within the five d orbitals. However, with increasing number of d electrons the procedure becomes extremely tedious, especially for configurations d^{3}–d^{7}. The manageable d^{2} case is exemplified in several textbooks (e.g. Schläfer & Gliemann, 1967) and will not be repeated here. Table 5 summarises the resulting freeion terms for all d^{N} configurations together with their total degeneracy (deg_{tot}). The ground term is the respective leftmost one, and terms with maximum spin multiplicity (conforming to Hund's rule) are indicated in bold. Since each electron configuration d^{N} and its holeequivalent d^{10–}^{N}^{} comprise the same terms, both such configurations are subject to the same type of interelectron repulsion. The relative term energies differ for various nd^{N}^{} configurations and can be expressed by two electrostatic parameters, for which the most popular notation is the socalled Racah parameters B and C.
Term splittings in crystal fields of different symmetry
Now that for free transition metal ions we have outlined the splitting into spectroscopic terms due to the interelectron repulsion H_{ee}, we are able to consider in details the effect of crystal field, acting as major perturbation H_{CF} to the freeion states ^{2}^{S}^{+1}L. Note that due to its electrical nature, crystal field acts only on the orbital parts of the wavefunctions and not on the spin parts. Using the group theory methods one can predict how the spectroscopic ^{2}^{S}^{+1}L terms split in CF of a given symmetry and then use the irreducible representations Γ_{α} (see e.g. Fig. 4) to label the CF levels (cf. also Part II and Beran et al., 2004 in this book). For illustration in Table 6 we list the CF levels arising after splitting of the respective L terms by a CF of cubic symmetry.
d^{N}, d^{10}^{N}  Freeion spectroscopic terms ^{2}^{S}^{+1}L  deg_{tot} 

d^{1}, d^{9}  ^{2}D  10 
d^{2}, d^{8}  ^{3}F, ^{3}P, ^{1}G, ^{1}D, ^{1}S  45 
d^{3}, d^{7}  ^{4}F, ^{4}P, ^{2}H, ^{2}G, ^{2}F, ^{2}D_{a}, ^{2}D_{b}, ^{2}P  120 
d^{4}, d^{6}  ^{5}D, ^{3}H, ^{3}G, ^{3}F_{a}, ^{3}F_{b}, ^{3}D, ^{3}P_{a}, ^{3}P_{b}, ^{1}I, ^{1}G_{a}, ^{1}G_{b}, ^{1}F, ^{1}D_{a}, ^{1}D_{b}, ^{1}S_{a}, ^{1}S_{b}  210 
d^{5}  ^{6}S, ^{4}G, ^{4}F, ^{4}D, ^{4}P, ^{2}I, ^{2}H, ^{2}G_{a}, ^{2}G_{b}, ^{2}F_{a}, ^{2}F_{b}, ^{2}D_{a}, ^{2}D_{b}, ^{2}D_{c}, ^{2}P, ^{2}S  252 
d^{N}, d^{10}^{N}  Freeion spectroscopic terms ^{2}^{S}^{+1}L  deg_{tot} 

d^{1}, d^{9}  ^{2}D  10 
d^{2}, d^{8}  ^{3}F, ^{3}P, ^{1}G, ^{1}D, ^{1}S  45 
d^{3}, d^{7}  ^{4}F, ^{4}P, ^{2}H, ^{2}G, ^{2}F, ^{2}D_{a}, ^{2}D_{b}, ^{2}P  120 
d^{4}, d^{6}  ^{5}D, ^{3}H, ^{3}G, ^{3}F_{a}, ^{3}F_{b}, ^{3}D, ^{3}P_{a}, ^{3}P_{b}, ^{1}I, ^{1}G_{a}, ^{1}G_{b}, ^{1}F, ^{1}D_{a}, ^{1}D_{b}, ^{1}S_{a}, ^{1}S_{b}  210 
d^{5}  ^{6}S, ^{4}G, ^{4}F, ^{4}D, ^{4}P, ^{2}I, ^{2}H, ^{2}G_{a}, ^{2}G_{b}, ^{2}F_{a}, ^{2}F_{b}, ^{2}D_{a}, ^{2}D_{b}, ^{2}D_{c}, ^{2}P, ^{2}S  252 
Term (L)  CF states 

S  A_{1g} 
P  T_{1g} 
D  E_{g}, T_{2g} 
F  A_{2g}, T_{1g}, T_{2g} 
G  A_{1g}, E_{g}, T_{1g}, T_{2g} 
H  E_{g}, T_{1g}, T_{1g}, T_{2g} 
I  A_{1g}, A_{2g}, E_{g}, T_{1g}, T_{2g}, T_{2g} 
Term (L)  CF states 

S  A_{1g} 
P  T_{1g} 
D  E_{g}, T_{2g} 
F  A_{2g}, T_{1g}, T_{2g} 
G  A_{1g}, E_{g}, T_{1g}, T_{2g} 
H  E_{g}, T_{1g}, T_{1g}, T_{2g} 
I  A_{1g}, A_{2g}, E_{g}, T_{1g}, T_{2g}, T_{2g} 
To provide an overview for all d^{N} ions, Figure 5 schematically depicts the CF splitting of the ground terms in octahedral O_{h} symmetry. In agreement with the oneelectron considerations in the previous chapter, in a tetrahedral CF of T_{d} symmetry the sequence of the respective CF levels is inverted and the g index is dropped (see the corresponding configurations in Fig. 5 in brackets). Upon symmetry reduction, the degenerate CF states (T_{1(g)}), T_{2(g)}, E_{(g)}) in Table 6 (and Fig. 5) split further according to the predictions of group theory. Selected cases of lower symmetry will be exemplified later.
On the basis of Tables 4 and 5, and Figure 5, which apply for the intermediate CF scheme, the following qualitative relationships between the term schemes and/or the ground term CF splittings arising for various d^{N} configurations can be established:
 The complete term and CF level scheme for the d^{N} ions at octahedral sites correspond to those for the d^{10–}^{N}^{} ions at tetrahedral and cube sites and vice versa.Fig. 5.Fig. 5.
The term schemes for the d^{N} ions at octahedral sites correspond to those for the d^{10–}^{N}^{} ions at octahedral sites, but the ground term CF split level sequence is inverted.
The d^{N} and d^{N}^{±5} ions at octahedral sites have an analogous CF splitting of the ground term, but different term schemes and spin multiplicities.
The d^{N} and d^{5–}^{N}^{} ions at octahedral sites as well as the d^{5+}^{N}^{} and d^{10–}^{N}^{} ions at octahedral sites have analogous but inverted CF splittings of the ground term, with different term schemes and spin multiplicities.
The relationships expressed in points 2 to 4 apply to tetrahedral and cube crystal fields as well.
Hence, if we limit our considerations to the CF levels arising from terms with the maximum spin multiplicity, just three fundamentally different cases can be discerned: (1) the d^{5} configuration with the S ground term, which does not split in a CF, (2) a group comprising d^{1}, d^{4}, d^{6} and d^{9} configurations with the D ground term, and (3) a group comprising d^{2}, d^{3}, d^{7} and d^{8} with the F ground term and an excited P term of the highest spin multiplicity. Figure 6 presents the socalled “Orgel diagrams”, which illustrate the dependence of the CF levels on the CF parameter Dq in octahedral and tetrahedral fields for the latter two cases. The CF energy levels arising from the D ground term in d^{1}, d^{4}, d^{6} and d^{9} configurations vary linearly with the CF strength and obey the centre of gravity rule, thus complying with the expectations arising from the oneelectron view as depicted in Figure 2. On the other hand, some levels evolving from the F and P terms with the maximum spin multiplicity for d^{2}, d^{3}, d^{7} and d^{8} configurations do not vary linearly with Dq and thus apparently violate the centre of gravity rule (indicated by the dotted lines). This is attributed to the “term interaction”, i.e. the mixing of the CF states described by the same representation, while arising from different terms with the same spin multiplicity. State mixing results in a “repulsion” of the energy levels, which increases with decreasing energy level gap, thus preventing the affected levels to cross in energy diagrams (“noncrossing rule”).
TanabeSugano diagrams
The TanabeSugano (TS) diagrams enable a more comprehensive presentation of the behaviour of energy levels of a transition metal ion in a CF versus its strength, as well as an easier correlation of experimental spectra with the calculated energies. They were first used by Tanabe & Sugano (1954) in their classical paper on the energy level calculations for various d^{N} configurations in cubic crystal fields. Generally such diagrams include, apart from the CF levels arising from the ground term, some relevant levels arising from the excited terms with lower spin multiplicity. Diagrams for cubic symmetry can be found in several textbooks. More comprehensive sets of the CF diagrams including tetragonal and trigonal symmetry cases have been published by König & Kremer (1977). TS diagrams show the CF energy level versus the cubic CF strength Dq, either in the energy units [cm^{−1}] or dimensionless units of Dq/B, where B is the Racah parameter. As a special feature, the ground state is plotted as the horizontal base line and the energies of all the excited levels refer to this base line. This facilitates data analysis since only relative energies are measured by optical spectroscopy (referring to the respective d^{N} ground state as zero), and not the absolute energy values. For illustration Figure 7 presents TS diagrams for d^{3} and d^{7} systems in octahedral coordination. The d^{3} configuration shows smoothly curved or linear energy changes with Dq. The energy levels in d^{7} exhibit an “apparent” discontinuity, indicated by a dashed vertical line, caused by a change of the ground state (i.e., forming the new baseline). This corresponds to a highspin to lowspin transition, which is expected for d^{4}–d^{7} configurations in strong crystal fields. The dotted vertical lines in Figure 7a and 7b represent roughly the situations found for Cr^{3}^{+}(d^{3}) and Co^{2+}(d^{7}) ions in minerals or related compounds with sixfold oxygen coordination.
An example illustrating correlation of experimental spectra with the CF levels represented by the TanabeSugano diagrams is shown in Figure 8 for a particular transition metal polyhedron. Such theoretically calculated diagrams can also be very useful in presenting the dependence of the energy levels on other lower symmetry CF parameters as well as the energy level changes upon polyhedral distortion or transition between different types of polyhedra. Practical examples are shown in Figure 9. Various distortion parameters involved in such cases are discussed in the following section.
Descent in symmetry and crystal field distortion parameters
It was mentioned already in the discussion of splittings of oneelectron energy levels that a single overall cubic field parameter Δ_{i} is not sufficient to describe the energy level splittings occurring due to polyhedral distortion and symmetry reduction from cubic O_{h} or T_{d}. As with the oneelectron d orbitals e and t, the multielectron degenerate levels T and E successively split into the nondegenerate levels A or B as follows from group theoretical predictions (see e.g. Cotton, 1990). For a complete description of crystal fields with arbitrary low symmetries (point groups C_{1}, C_{i}) up to fourteen “real” and “imaginary” CF parameters are needed (see e.g. Rudowicz, 1986a; Yeung & Rudowicz, 1992; Chang et al., 1994). Such cases can only be handled by applying sophisticated semiempirical calculation methods discussed in Part II.
For axial symmetry distortions described by the tetragonal (D_{4}_{h}, D_{4}, C_{4}_{v}, D_{2d}) and trigonal (D_{3}_{d}, D_{3}, C_{3v}) type I point groups three CF parameters are sufficient, whereas for the axial type II cases one more CF parameter is needed (Rudowicz, 1985a, 1986b). Various CF “distortion” parameters have been widely used in the chemical and physical literature, resulting in an awkward situation, which have been succinctly summarised by Gerloch & Slade (1973): “The parameters used vary from coordination number to coordination number, from symmetry to symmetry and from author to author. It is not always clear whether any or all of these parameters are, or can be, related to one another. Nor is it clear how the parameter values deduced from the spectra or magnetism of ‘distorted’ systems reflect geometrical distortions as opposed to some radial properties which may be related to bonding in some way analogous to the behavior of 10 Dq.”
Two notations for general forms of CF Hamiltonians (see Part II) based on the tensor operators are now prevailing in the literature, namely: (i) the Wybourne operators (Wybourne, 1965; Rudowicz, 1987) and the extended Stevens (ES) operators (Rudowicz, 1985b, 1987). For historical reasons, the conventional CF parameters, which to a certain extent are simply related to noncubic distortions, are still widely used in the chemical literature, in particular the parameters Dq_{tetr}, Dt, and Ds for tetragonal cases and Dq_{trig}, Dτ, and Dσ for trigonal ones (see e.g. Ballhausen 1962; Yeung & Rudowicz, 1992). Dt and Dτ represent the fourthorder CF terms (see Part II) and contain direct geometrical information, while Ds and Dσ represent the secondorder CF terms and are not related to geometrical distortions in a straight forward way. Note also that Dq, as indicated by the subscripts, no longer represents a “cubic” quantity nor reflects an “averaged” CF strength (mostly denoted by Dq_{cub}). Dq_{tetr} and Dq_{trig} quantify the equatorial CF strength only, i.e., the CF component along the axes perpendicular to the four and threefold axis, respectively. Unfortunately, the symbols used in the literature are often not clearly defined, thus leading to ambiguities in their interpretation. The averaged cubic Dq values are given in the tetragonal case by Dq_{cub} = Dq_{tetr} – 7Dt/12, whereas in the trigonal case by Dq_{cub} = Dq_{trig} + 7Dτ/18. Hence, positive values of Dt correspond to an octahedral elongation along the fourfold axis, whereas the negative values to an octahedral compression (see Fig. 9b). Assuming the validity of the 1/R^{5} dependence of the CF strength with the metalligand bond length (see above), the tetragonal elongation or compression of an octahedron along the fourfold axis can be calculated from Dq_{tetr} and Dt as the percentage distortion (with respect to the equatorial M–L distances): [(100/(1 – (Dt/4Dq_{tetr}/7)^{1/5}) – 100](%). In the trigonal case, the positive values for Dτ also correspond to an octahedral elongation along the threefold axis, whereas the negative values to a trigonal compression (see Fig. 9a; note the nonlinear behaviour of Dτ). The polar angle θ between the M–L bonds and the threefold axis (θ is 54.74° in an ideal octahedron) is related to Dτ as Dτ = (35 cos^{4} θ – 30 cos^{2} θ + 3))/21 cosθsin^{3}θ] – 2 Dq_{trig}/3. Note that the analogous sign convention for Dt and Dτ, relating positive values to an octahedral elongation and negative values to an octahedral compression, is somewhat misleading since an octahedral elongation along a tetragonal axis decreases the axial CF strength as compared with the equatorial one, while an octahedral elongation along a threefold axis increases the axial CF strength and decreases the equatorial one. For compression, reversed relations are observed. This fact is reflected by the negative sign at Dt whereas positive sign at Dτ in the expressions defining Dq_{cub} (see above). Furthermore, the reader should be aware of the fact that, even for the wellknown distortion parameters, different sign conventions as well as other definitions of Dq_{cub} and Dq_{trig} occur in the literature. Several definitions of and relations between the latter parameters and numerous other less common CF parameters for lower symmetries have been compiled, e.g. by Gerloch & Slade (1973), Lever (1984), König & Kremer (1977), Morrison (1992) and Yeung & Rudowicz (1992). König & Kremer (1977) furthermore present about 400 complete TanabeSugano diagrams, calculated for tetragonal and trigonal symmetries (in terms of Dt, Ds, Dτ, Dσ) for all electron configurations from d^{2} to d^{8}.
Selection rules
As discussed above, the freeion d^{2} to d^{8} electron configurations split due to the electronelectron interactions into the ground and several excited spectroscopic ^{2}^{S}^{+1}L terms with total degeneracy as listed in Table 5. Table 6 shows how these terms split further in a CF of cubic symmetry. Lower symmetry CF (e.g. Fig. 9) as well as the spinorbit (SO) coupling (see below) lead to further splittings, resulting in a large number of discrete energy levels. Note that for configurations with an odd number of d electrons (the socalled Kramers ions with halfinteger total spin), their levels consist of “Kramers doublets”, which remain degenerate in the absence of a static magnetic field B, whereas for the Kramers ions with even N and hence integer total spin, the Kramers doublets may be split by the combined action of CF and SO interactions even in B = 0. Consequently, one might expect that an incident electromagnetic radiation of the visible and neighbouring ranges is completely absorbed and that minerals and related compounds containing respective transition elements as major components appear opaque. This is evidently not the case due to low or vanishing probability (and thus intensity) of transitions from the ground level to most of the excited energy levels. These transitions are governed by a number of quantum mechanical selection rules discussed below.
The intensity of the relevant electric dipole transitions (for our purposes, we can neglect the magnetic dipole and electric quadrupole transitions) depends on the extent of charge redistribution during the transition, i.e., the change of the dipole moment μ. Hence the intensity is proportional to the transition moment Q defined as
where Ψ_{a} and Ψ_{b} are the CF wavefunctions of the ground state and the excited state involved in the transition under consideration, respectively, and M is the dipole moment operator of the electromagnetic radiation. Unfortunately, detailed quantitative calculations of Q have, till recently, been hardly feasible. Nevertheless, instead of the absolute transition intensities we can gain limited information on which transitions are possible. Group theory enables to predict the nonzero matrix elements in Equation 17. Note that M transforms as a vector, i.e., according to the D^{(1)} representation of the full rotation group, which decomposes in a given symmetry CF into ΣΓ_{α} in a similar way as the p orbitals (Fig. 1). If the irreducible representations Γ_{a} of Ψ_{a}, and Γ_{b} of Ψ_{b} are known (or assumed), then the following theorem enables predictions of the nonzero matrix elements: An electronic dipole transition is allowed if the group theoretical product Γ_{a} × (ΣΓ_{α}) × Γ_{b} after its decomposition into the irreducible representations contains the totally symmetric representation A_{1g} at least once.A_{1g} is called the totally symmetric representation, since it represents all properties of a system which are invariant under all symmetry operations of a given point symmetry group. Since M has the same symmetrical properties of a vector, in an octahedral field it transforms as T_{1u} (i.e., ΣΓ_{α} ≡ T_{1}_{u} only). The above theorem and group theoretical methods (e.g. Cotton, 1990) provide an explanation of the first selection rule, originally based on experimental data:
Laporte (or parity) selection rule: Δl = ± 1 (Δl ≠ 0)
The Laporte selection rule states that the electric dipole transitions may occur only between states differing in their orbital quantum number l by one, i.e., s → p, p → d, d → f transitions (and vice versa) are allowed. For example, LMCT and MLCT transitions occurring as “absorption edges” in the near UV region (sometimes extending into the visible range) are allowed and hence very intense. On the other hand, the Laporte rule implies that transitions within orbitals of the same character are forbidden. This applies also to the d →d (or f →f) transitions we are dealing with in the context of the CF formalism. For the d orbitals with even parity (gerade, g) this rule is easily verified by considering the product , which yields the subscripts “product” g × u ×g = u (ungerade), and hence cannot contain A_{1g}. However, the Laporte rule can be relaxed to a certain extent by quantum mechanical admixture of some (acentric) p orbitals into the d orbitals. This p–d mixing is only possible if the respective transition metal complex lacks a centre of symmetry. Then the g and u indices are removed from the representations (as outlined above for the tetrahedral case) and consequently the product may now contain A_{1}. The d–d transition probability and hence the absorption intensity is related to the degree of “acentricity”, i.e. for a particular transition ion hosted in comparable chemical environments the following sequence of the absorption intensity may be observed: tetrahedral fields >> strongly acentric distorted octahedral fields > weakly distorted or “pseudocentric” octahedral fields > centric (octahedral) fields. Figure 10 exemplifies this relationship for several synthetic Co^{2+} selenites. It provides comparison between the (averaged polarised) spectrum of CoO_{4} tetrahedra (point symmetry S_{4}) in Ca_{3}Co(SeO_{3})_{4} (Wildner, 1996a) and those of trigonally distorted CoO_{6} octahedra in three closely related KCo selenites (Wildner & Langer, 1994a). In zemannitetype K_{2}Co_{2}(SeO_{3})_{3}·2H_{2}O the octahedra are strongly acentric and distorted with point symmetry C_{3}; the integral absorption of the intense ^{4}T_{1(g)} (cubic) band system in the visible spectral range is onethird of the CoO_{4} tetrahedron. In K_{2}Co_{2}(SeO_{3})_{3} the octahedra exhibit a weaker bond angle distortion and acentricity with symmetry C_{3}_{v} and the integral intensity is 20%; finally, buetschliitetype K_{2}Co(SeO_{3})_{2} contains centric octahedra with symmetry D_{3}_{d}, and the ^{4}T_{1(g)} band is only 5% of that for the tetrahedron. It is worth mentioning that Co^{2+} in tetrahedral coordination is one of the strongest chromophores among the d transition elements.
In the latter case of a centric CF, the Laporte rule applies strictly and can only be relaxed dynamically due to acentric polyhedral vibrations. In the octahedral case, three out of six fundamental vibrational modes are acentric (T_{2}_{u} and 2 × T_{1}_{u}) and thus violate the inversion symmetry during vibration, resulting in a small transition probability even for structurally centric positions. In other words, we can introduce such an acentric octahedral mode into the product Γ_{a}(Ψ_{a}) × Γ_{α}(M) × Γ_{b}(Ψ_{b}) × Γ_{γ}(Ψ_{vib}) implying g × u × g × u = g, which may contain A_{1g}. This situation is called “vibronic coupling” and the respective transitions are said to be “vibronically allowed”. Table 7 summarises ranges of the absorption intensities for the d–d and other electronic transitions.
From these considerations it follows immediately that the influence of temperature on the intensities of absorption bands is much more pronounced for octahedral and centric crystal fields (e.g. octahedral complexes) than for acentric ones (e.g. tetrahedral complexes). Upon cooling, the magnitude of molecular vibrations decreases leading to a strongly reduced absorption intensity for vibronically allowed transitions. As an example, the temperaturedependent polarised absorption spectra of trigonally compressed Co(OH)_{6} octahedra with point symmetry D_{3}_{d} in brucitetype Co(OH)_{2} are shown in Figure 11 (Andrut & Wildner, 2001). Contrarily, in tetrahedral fields the contribution of polyhedral vibrations to the transition probability is marginal and hence only a slight temperature dependence of the absorption intensity is observed.
Spin selection rule: Δ(2S + 1) = 0
The spin selection rule states that the transitions between states with different spin multiplicity 2S + 1 are forbidden. Since, in the most common case of the highspin configurations, the ground state emerges from the freeion term with the highest spin and orbital multiplicity, only transitions between the levels arising from this ground term (this applies to d^{1–4,6–9} configurations) and to the levels arising from the only excited term with the same highest spin multiplicity (this applies only to d^{2,3,7,8} configurations) are spinallowed. The ^{6}S ground term of d^{5} systems is orbitally nondegenerate and does not split in crystal fields. Hence, for the highspin d^{5} systems all transitions are spinforbidden, leading to the comparatively light colour of many Mn^{2+} and Fe^{3+} minerals and synthetic compounds (provided that MMCT of Fe^{2+}/Fe^{3+} are absent).
Spinforbidden transitions may nevertheless gain some intensity (see Table 7) due to the effect of the spinorbit coupling.
Spinorbit coupling and “iniensity stealing”
In the context of the RussellSaunders (LS) coupling scheme discussed above, it is assumed that the spin momenta of all d electrons and their orbital momenta couple among themselves leading to ^{2}^{S+}^{1}L freeion terms with the spin (2S + 1) and orbital (2L + 1) multiplicity. However, with the increasing atomic number Z of the transition metal ions this assumption is less strictly fulfilled as the single electron spin and orbital momenta also couple appreciably among each other due to the stronger spinorbit coupling (parameterised by the SO coupling constant ξ). This leads to further splittings of the energy levels and relaxation, to a certain extent, of the spin selection rule. The resulting admixture of the spinallowed character to the spinforbidden states and hence their intensity enhancement especially increases in the vicinity of the spinallowed or Laporteallowed (absorption edges) transitions. This effect is also called “intensity stealing” or “intensity borrowing” (e.g. Lever, 1984). Figure 12 shows, as an example, the spinforbidden transitions ^{4}A_{2g}(F) → ^{2}E_{g}(G) and → ^{2}T_{1g}(G) of Cr^{3+} (the socalled “ruby lines”) in the spectra of two modifications of Cr(SeO_{2}OH)(Se_{2}O_{5}) (Wildner & Andrut, 1998). In modification I the energy gap between the ^{2}E_{g}(G) level and the spinallowed ^{4}T_{2g}(F) band is approximately half of that in modification II. The corresponding quartet admixtures to ^{2}E_{g}(G) are calculated as ∼ 2% in I and ∼ 0.6% in II. The transition to ^{2}T_{1g}(G) is generally weaker and is only observed in I (∼ 2% quartet character, while < < 0.3% in II). Theoretically, at the crossing point between the spinallowed and spinforbidden energy levels, these transitions cannot be distinguished any longer. The effect of spinorbit coupling and hence intensity stealing is rather weak for the 3d^{N} series, and increases on going to the 4d^{N} and 5d^{N} elements (Eqn. 16a,b). Finally, for the f^{N} transition ions, the LS coupling scheme is no longer applicable and splittings due to the SO coupling are stronger than those due to CF and electronelectron interactions (Eqn. 16c). This leads to quite different spectra of lanthanide and actinide complexes. Values of ξ for the d^{N} and f^{N} series are compiled, e.g., by Figgis & Hitchman (2000).
Electronic selection rule: Δe^{−} = 1
According to this rule, transitions representing a change in the electron configuration involving two electrons are forbidden. As a result, transition probabilities and intensities of such “twoelectron jumps” are much lower than expected from other selection rules. The polarised spectra of K_{2}Co_{2}(SeO_{3})_{3} in Figure 8 can be considered as a representative example for the relative intensities of the three spinallowed d–d bands of Co^{2+}O_{6} octahedra, i.e., in O_{h} symmetry ^{4}T_{1g}(^{4}F) [ground state] → ^{4}T_{2g}(^{4}F), → ^{4}A_{2g}(^{4}F) and → ^{4}T_{1g}(^{4}P). In the strong field approximation, the ^{4}T_{1g}(^{4}F) → ^{4}A_{2g}(^{4}F) transition observed around 15 000 cm^{−1} corresponds to a twoelectron jump resulting in the weak intensity of this band.
Selection rules arising from symmetry considerations
These selection rules deal with the orientational dependence of the electric dipole transition intensities in noncubic crystals when linearly polarised light is applied. In this way the pleochroic behaviour observed in many minerals and related compounds containing transition metal ions can be elucidated using group theoretical methods. For that purpose, the group theoretical multiplication Γ_{a}(Ψ_{a}) × Γ_{α}(M) × Γ_{b}(Ψ_{b}) [× Γ_{γ}(Ψ_{vib})] has to be carried out in detail and the product checked for the presence of the A_{1g} representation (or A_{1} or A, depending on the actual CF symmetry). A worked example (Cr^{3+} at an acentric trigonal site) is given below. Similar examples, some of which deal with the Γ_{γ}(Ψ_{vib}) contribution for centrosymmetric sites, can be found in several of the relevant textbooks cited. Note that the intense MMCT transitions, which are strongly polarised along the M–M vector, cannot be treated by these selection rules.
Proper identification of symmetry selection rules poses sometimes problems, which are exemplified in the following.
• Symmetry of the crystal field, i.e., the local site symmetry and the type of coordination polyhedron centred at the transition ion.
Here, two extreme situations have to be distinguished. (1) The transition ion exclusively occupies a specific structural site in an endmember or synthetic crystal. Then, the point symmetry, the type of the coordination polyhedron, its distortion, and the actual bond lengths and angles (the latter are necessary for the quantitative energy level calculations discussed in Part II) can be exactly determined using single crystal diffraction methods. (2) The transition metal ion is only a minor or trace component, occasionally occupying available structural sites or interstitials as point defects. In this case, the structure studies using crystalaveraging diffraction experiments can yield only very limited or no information at all, depending on the relative concentration of these point defects. Then the most probable structural site for incorporation has to be guessed by crystal chemical arguments, by the results of an optical absorption experiment, or by other nonaveraging spectroscopic methods. However, detailed structural information (i.e., bond lengths and angles) can hardly be obtained using these methods (but see Part II).
• Descent in symmetry.
Once the point symmetry is established, the corresponding representations Γ_{α}(M), Γ_{a}(Ψ_{a}), Γ_{b}(Ψ_{b}), Γ_{γ}(Ψ_{vib}) have to be determined. The components of the dipole moment operator M transform according to D^{(1)} representation, i.e., in the same way as those of a vector (x, y, z), and hence the irreducible representations Γ_{α} can be easily found in the respective Character Tables. The representations suitable for Ψ_{a}, Ψ_{b} (and Ψ_{vib}) can be obtained using the method of “descent in symmetry” (e.g. Cotton, 1990), i.e., a gradual reduction of symmetry. Starting with the representations arising for the cubic polyhedral symmetry (O_{h} for octahedra and cubes, T_{d} for tetrahedra), the group theoretical “Correlation Tables” show how the higher symmetry representations transform and split upon the stepwise reduction of symmetry to the actual point symmetry group. This has to be done with caution, taking into account the intricacies pertinent especially to orthorhombic and monoclinic point groups. In particular, the exact type of the symmetry elements prevailing in the lower symmetry point group has to be ascertained, e.g., horizontal, vertical, or diagonal mirror planes (σ_{h}, σ_{v}, σ_{d}), or the respective twofold axes (C_{2}, C_{2}', C_{2}”). Wrong interpretations of the CF absorption spectra arising from such intricacies are occasionally found in the literature. A detailed example is given by Goldman & Rossman (1977). Correlation Tables can be found in most textbooks on group theory, particularly extensive and useful are those compiled by Ferraro & Ziomek (1975).
• Determination of the ground state energy level.
If the ground state of a d^{N} system is nondegenerate even in cubic symmetry (i.e. for d^{3,5,8} systems) and hence is not split in a CF of lower symmetry, its corresponding representation is easily obtained by following its transformations in the Correlation Tables. Levels degenerate in cubic symmetry (i.e. for d^{1,2,4,6,7,9} systems) split in lower symmetry crystal fields, giving rise to two or three possible new ground states. The actual ground state can be elucidated by one of the following methods:
Trial and error: perform the group theoretical analysis of the symmetry selection rules for all possible ground states and compare the results with experiment.
Deduction: in higher symmetry systems with distinct polyhedral distortions (e.g. polyhedral elongation or compression along a fourfold or threefold symmetry axis) the ground states are mostly wellestablished and can be easily deduced or taken from literature.
Calculation: energy level calculations based on semiempirical methods (see Part II) can yield the ground state for an arbitrary low symmetry and the arising distortions.
• Consideration of pseudosymmetry.
The crystal structures of many geoscientifically important minerals and related synthetic compounds often provide only lowsymmetry structural sites suitable for the incorporation of transition metal ions. As long as we do not apply semiempirical methods to carry out the energy level calculations (see Part II), such low point symmetry cases are often hardly amenable for interpretation of the observed polarisation. Particularly, in triclinic point groups all CF states finally transform as A (C_{1}) or A_{g} (C_{i}), and thus all transitions are allowed, at least vibronically. For monoclinic and orthorhombic sites, a detailed group theoretical analysis is feasible, but sometimes it does not reflect a distinct polarisation behaviour properly. In these cases the respective coordination polyhedron should be checked for pseudosymmetry, i.e., an approximate symmetry higher than the actual crystallographic one. The polarisation effects should be then reanalysed with respect to the selection rules for the higher point symmetry group. This method is also called “ascent in symmetry”.
• Polyhedral orientation.
Polyhedra with low point symmetry occurring in crystals of higher symmetry as well as crystals with low symmetry in general pose further problems. In the former case, such crystallographically equivalent polyhedra often occur with different orientations. This leads to mixed contributions from electronic transitions induced along two or three polyhedral axes oriented differently with respect to the polarisation direction, which is restricted by the optical indicatrix of high symmetry. The contributions mix according to a cos^{2}θ dependency. If a polyhedron adopts parallel orientations in low symmetry crystals but the polyhedral axes do not coincide with the axes of the optical indicatrix, then the best possible information can be obtained by measuring at most six polarised spectra in three orthogonal sections of the crystal. These procedures have been discussed by Dowty (1978), Libowitzky & Rossman (1996), and Hitchman & Riley (1999).
A worked example: Al_{2}O_{3}:Cr^{3+} (ruby)
• Structure and symmetry:
Ruby is a gemstone variety of corundum, αAl_{2}O_{3}. It crystallises in the trigonal space group with hexagonal axes a = 4.761 Å and c = 12.995 Å. The Al cations occupy position 12c (0, 0, z ∼ 0.35) with point symmetry 3 (C_{3}), the oxygen atoms reside on position 18e (x ∼ 0.31, 0, ¼), point symmetry 2 (C_{2}). Al–O distances are 1.856 Å and 1.971 Å – each three times (Finger & Hazen, 1978). In natural ruby crystals, Cr^{3+} cations (up to 2 wt% Cr_{2}O_{3}) are incorporated into the Al_{2}O_{3} structure, giving rise to the beautiful rubyred colour of this gemstone. Crystal chemical considerations and spectroscopic results unequivocally show that Cr^{3+} replaces Al^{3+} cations, thus forming CrO_{6} polyhedra with point symmetry 3. However, since the Cr^{3+} cation is larger than Al^{3+}, the local environment of the Cr^{3+} defect will relax to some extent; the detailed bond lengths and angles of the resulting CrO_{6} polyhedron could not be established so far. This problem will be dealt with in details in Part II.
• Freeion and crystal field states and the energy level splitting.
Cr^{3+} has 3d^{3} electron configuration with a quartet ^{4}F ground term and the excited ^{4}P, ^{2}G, ^{2}H and higher doublet terms. In a cubic crystal field ^{4}F splits into ^{4}A_{2}_{g} (ground state), ^{4}T_{2g}, and ^{4}T_{1g} (see Fig. 5). Since the ground state is nondegenerate, no further splitting occurs upon symmetry reduction. Table 6 shows how the other terms split in O_{h} symmetry. In a trigonal field of C_{3} symmetry, the cubic CF levels split or transform as listed in the partial Correlation Table (Table 8).
O_{h}  D_{3}_{d}  C_{3}_{v}  C_{3} 

A_{1g}  A_{1g}  A_{1}  A 
A_{2g}  A_{2g}  A_{2}  A 
E_{g}  E_{g}  E  E 
T_{1g}  A_{2g} +E_{g}  A_{2} + E  A + E 
T_{2g}  A_{1g} + E_{g}  A_{1} + E  A + E 
O_{h}  D_{3}_{d}  C_{3}_{v}  C_{3} 

A_{1g}  A_{1g}  A_{1}  A 
A_{2g}  A_{2g}  A_{2}  A 
E_{g}  E_{g}  E  E 
T_{1g}  A_{2g} +E_{g}  A_{2} + E  A + E 
T_{2g}  A_{1g} + E_{g}  A_{1} + E  A + E 
• Character Table for C_{3}:
For illustration we reproduce the Character Table for the point symmetry group C_{3} in Table 9 (modified from Cotton, 1990). For the present purpose, only the characters of the components of the dipole moment operator are of interest, i.e., M_{z} parallel to the threefold z axis transforms as A, while M_{x} and M_{y} perpendicular to the threefold axis transform as E.
C_{3}  E  C_{3}  Vector components (e.g. translations, dipole moment operator M, p orbitals and rotations R_{i})  Tensor components (e.g. d orbitals)  

A  1  1  1  z, R_{z}  x^{2}+y^{2}, z^{2} 
E  (x^{2}y^{2}, xy) (yz, xz) 
C_{3}  E  C_{3}  Vector components (e.g. translations, dipole moment operator M, p orbitals and rotations R_{i})  Tensor components (e.g. d orbitals)  

A  1  1  1  z, R_{z}  x^{2}+y^{2}, z^{2} 
E  (x^{2}y^{2}, xy) (yz, xz) 
• Calculation of group theoretical products Γ_{a}(Ψ_{a}) × Γ_{α}(M) × Γ_{b}(Ψ_{b}) × Γ_{γ}(Ψ_{vib}):
Since the point group under consideration (C_{3}) is acentric, we can here omit Γ_{γ}(Ψ_{vib}) from calculations. Setting up the relevant multiplication tables (Table 10) immediately shows which of the triple products: Γ_{a}(Ψ_{a}) × Γ_{α}(M_{i}) × Γ_{b}(Ψ_{b}) × Γ_{γ}(Ψ_{vib}) equals or contains the totally symmetric representation, i.e., A in the case of C_{3}. This leads to the selection rules for transitions between CF states given in Table 11, i.e., the A → A transitions are allowed when the linearly polarised electric light vector is parallel to the threefold axis, A → E and E → A transitions are allowed if this vector is perpendicular to this axis, and E → E transitions are allowed for both orientations.
• Interpretation of the polarised absorption spectra.
The polarised absorption spectra of a synthetic ruby crystal shown in Figure 13 are characterised by two intense bands at 18 000 and 24 500 cm^{−1}, which are typical for Cr^{3+} in octahedral oxygen coordination exhibiting a strong CF like in the matrix Al_{2}O_{3}. These bands with typical full widths at half maximum (FWHM) of 2500 cm^{−1} and 3500 cm^{−1} are correlated to the spinallowed d–d transitions ^{4}A_{2g}(^{4}F) → ^{4}T_{2g}(^{4}F) and ^{4}A_{2g}(^{4}F) → ^{4}T_{1g}(^{4}F) in O_{h} symmetry, respectively. The third spinallowed transition ^{4}A_{2g}(^{4}F) → ^{4}T_{1g}(^{4}P), expected at around 39 000 cm^{−1}, could not be measured. The spectra show a significant pleochroism with energy splittings in the order of 500 cm^{−1}, indicating a substantial trigonal perturbation of the octahedral CF. According to the symmetry selection rules for point group C_{3}, the trigonal A → E components occur in polarisation perpendicular to the threefold axis at 17 890 and 24 340 cm^{−1}, the trigonal A → A transitions in polarisation parallel to the C_{3} axis at 18 360 and 25 120 cm^{−1}. An analysis of these spinallowed bands yields the following parameters: Dq_{trig} = 1858 cm^{−1}, Dτ = –132 cm^{−1}, Dσ = 5 cm^{−1}, and Racah B_{35} = 632 cm^{−1} (the subscript 35 refers to the Racah parameter derived from the spinallowed bands, cf. section 3.6.).
In addition to the spinallowed bands, several spinforbidden quartet → doublet transitions are clearly observed in the optical spectra. The socalled “ruby lines” R_{1} and R_{2} with FWHM ≤ 10 cm^{−1} are located at 14 399 and 14 428 cm^{−1}, respectively, and correspond to the transitions between states ^{4}A_{2g}(^{4}F) → ^{2}E_{g}(^{2}G) split up due to spinorbit coupling. Components of ^{4}A_{2g}(^{4}F) → ^{2}T_{1g}(^{2}G) are found at 14 945 and 15 162 cm^{−1}, and those of ^{4}A_{2g}(^{4}F) → ^{2}T_{2g}(^{2}G) at 20 972, 21 035 and 21 332 cm^{−1}. In a first approximation, these spinforbidden transitions are nearly fieldindependent and hence correspond to more or less horizontal energy levels in the respective TanabeSugano diagram (Fig. 7). Further details of the assignment of absorption bands in ruby have been given, e.g., by McClure (1962) and MacFarlane (1963). In Part II we will further outline how information about the local geometry of the CrO_{6} polyhedron within the Al_{2}O_{3} matrix can be extracted from the polarised absorption spectra by applying the semiempirical superposition model (SPM) of crystal fields.
d–d absorption bands
Band widths – general aspects
As it is evident from the examples of the spectra discussed in earlier sections, absorptions due to the electronic d–d transitions generally give rise to rather broad bands with FWHH in the order of 1000–3000 cm^{−1}, whereas narrow bands (about 10–100 cm^{−1}) are observed only occasionally, e.g. in the spectra of ruby. If we consider at first a single absorption band due to a particular d–d transition in a welldefined transition metal complex, such differences can be rationalised with respect to the different slopes of the respective energy lines in TanabeSugano (TS) diagrams. On the one hand the atoms are subject to permanent vibrations and the metalligand bond lengths actually fluctuate around a mean value, which can be established e.g. by Xray diffraction methods. On the other hand, the crystal field splitting Δ is very sensitive to the M–L distances, theoretically according to the 1/R^{5} relationship, and hence Δ varies over a certain range during vibration. Consequently, the band widths may be correlated to the slope of energy levels in the TanabeSugano diagram, i.e., the energy lines with a high positive slope give rise to rather broad absorption bands, while narrow bands belong to more or less horizontal energy levels. The former case applies to all spinallowed and many spinforbidden states, the latter to certain spinforbidden states, namely those where an electron spin inversion occurs within a set of degenerate orbitals, e.g. within the t_{2g} orbitals. Such “spinflip” transitions are observed as rather sharp bands, for example in the spectra of Cr^{3+}(d^{3}) in ruby (Fig. 13) around 14 400, 15 000 and 21 000 cm^{−1}. The excited states as well as the ground state in this case belong to configuration and are thus hardly affected by the CF splitting Δ. Since the integral intensity of the spinforbidden transitions is generally rather small (compare Table 7), the majority of the fielddependent, i.e., broad spinforbidden transitions is mostly obscured by intense spinallowed bands or by the spectral background noise. Nonetheless, they can occasionally be observed, e.g. in d^{5} systems (e.g. Mn^{2+}), which lack the spinallowed bands (but see also Co^{2+} spectra in Fig. 8).
Influence of temperature and pressure
The behaviour of band widths and band positions with temperature can be interpreted along the general lines discussed above. At elevated temperatures both the polyhedral vibrations as well as the M–L bond lengths generally increase and the respective Δ range is expanded. Hence, for transitions with a positive slope in the TS diagram the band widths increase and bands will shift to lower wavenumbers (red shift). Upon cooling the inverse behaviour is observed, i.e., smaller band widths and a blue shift. Absorption bands arising from more or less horizontal energy levels in the TS diagram are much less affected by temperature changes. For comparatively weak crystal fields, e.g. tetrahedral ones, such effects are generally smaller.
The pressure dependence of crystal field spectra (e.g. Langer, 1990; Burns, 1982, 1993) can be roughly interpreted as inverse temperature dependence, since the M–L distances decrease upon increasing pressure. Hence, blue shifts of the absorption bands to higher wavenumbers will be usually observed. However, the band widths are not reduced significantly or may even tend to increase with pressure. Depending on the particular crystal structure and site symmetry, polyhedral compression moduli may be markedly anisotropic and hence splittings of band components may rather correlate more closely with pressure than with temperature.
FranckCondon principle and vibrational structure
The FranckCondon principle allows a more stringent interpretation of band widths as well as of the vibrational structure of an electronic absorption band, which may be eventually resolvable at low temperatures. The extent of charge redistribution accompanying an electronic transition depends on the particular type of transition. Spinallowed transitions will usually transfer charge into the antibonding orbitals. The resulting distorted charge distribution (as compared with the ground state) will stimulate molecular vibrations and shift the potential energy curve of the excited state to the higher M–L equilibrium distances R_{e}, as compared with the ground state distance R_{0}. However, according to the FrankCondon principle, the electronic transitions take place in a much shorter time (10^{−15}–10^{−18} s) than the molecular vibrations (10^{−10}–10^{−13} s). Therefore the vertical lines in the potential energy diagrams represent the former transitions as shown in Figure 14. On the other hand, during spinforbidden spinflip transitions the charge distribution remains approximately the same, and consequently the potential energy curve of the excited state corresponds to the same or a similar equilibrium distance as for the ground state. At ambient or lower temperatures, transitions usually occur from the vibrational ground level of the electronic ground state to one or more vibrational levels of the excited electronic state, depending on the equilibrium distance and temperature. The resulting band shapes are illustrated in Figure 14, which shows that for small differences of R_{e}vs. R_{0} asymmetric band shapes are expected due to vibronic progression. Usually, the inherent vibrational structure of such absorption bands can be detected only at low temperatures. The energy separations between the vibrational components are in the order of a few 100 cm^{−1}. As an example, Figure 15 shows the asymmetric spinforbidden ^{4}A_{2g}(F) → ^{2}E_{g}(G) and ^{4}A_{2g}(F) → ^{2}T_{1g}(G) bands of Mn^{4+}(3d^{3}) exhibiting distinct vibrational fine structure at reduced temperatures.
At elevated temperatures, higher vibrational levels of the electronic ground state may be populated and then electronic transitions may also arise from these levels. Consequently, the respective transition energies of such “hot bands” are reduced by a few hundreds of cm^{−1}, thus leading to a further red shift and the broadening of the absorption bands in addition to the respective effects caused by the increased M–L distances and molecular vibrations.
Apart from the basic principles discussed above, there are several other factors influencing band widths. These include, for example:
Level splitting in crystal fields of low symmetry: for small to moderate distortions, split levels might not be detectable as distinct bands or shoulders but only cause broadening of the respective “cubic” bands. In the special case of JahnTeller distortions (see above), systems with the cubic E ground states (octahedral d^{4} and d^{9}) are usually strongly and statically distorted and exhibit clearly discernible split components. However, also the cubic T ground states may be subject to (very) small – and then hidden – or dynamic distortions leading to increased band widths. Furthermore, the excited degenerate states (again especially E) may also be affected by JahnTeller distortions; a wellknown example is the ^{2}T_{2g} → ^{2}E_{g} transition of Ti^{3+}(3d^{1}) appearing as broadened asymmetric absorption band (e.g., Burns, 1993).
Splitting due to the spinorbit coupling: for the firstrow transition elements, the spinorbit splitting plays a minor role and can be detected only occasionally (e.g., for Co^{2+} in tetrahedral coordination); usually, only a band broadening by a few 100 cm^{−1} is observed.
Distribution of a transition ion on several structural sites: many crystal structures of minerals or related inorganic compounds provide two or more structural positions suitable for transition metal ions. If the occupied positions are similar in their geometrical and chemical environment, then their contributions to the absorption spectrum will, to a large extent, overlap and result in broad bands.
Order/disorder phenomena: orderdisorder transitions or variations in the degree of order in transition metal ion bearing solid solutions will affect band widths to some extent. Disordered phases or modifications generally exhibit the comparatively larger band widths.
One has to bear in mind that up to now changes in the CF interactions only were taken into account to explain band shifts and shapes under variable conditions (of temperature and pressure). As a matter of fact, the bonding character and hence the interelectronic repulsion in a transition metal complex may also be affected, thus resulting in changes of the Racah parameters B and C and concomitant band shifts. These aspects will be outlined in the next chapter.
Qualitative appraisal of the parameters Dq, B and C
The extraction of Dq from optical absorption spectra of octahedral d^{N} systems is straightforward only for the d^{1} and d^{9} configurations, where the interelectronic repulsion does not exist (i.e. no Racah parameters are involved) and hence the energy of the single absorption band corresponds directly to 10Dq. However, both d^{1} and especially d^{9} are liable to JahnTeller distortion and hence to the splitting of the excited energy level. In all the other d^{N} systems more than one absorption band is usually observed in the spectra, and hence the parameters often can be extracted in various ways, sometimes leading to slightly different results. Algebraic expressions for the most important energy levels of d^{2–8} configurations in terms of Dq, B, and C are given, e.g., in Lever (1968, 1984). Usually for d^{4} and d^{6}, 10Dq is also taken as the energy of the single spinallowed band since it is not affected by term interaction; if possible, B and C have to be extracted from the observed spinforbidden transitions. In the manyelectron cases d^{2,3,7,8} three spinallowed bands (v_{1}–v_{3}) may be observed enabling the determination of Dq and Racah B. As an approximation, the energy difference between A_{2g} and T_{2g} levels arising from the F ground term can be taken as 10Dq. For d^{3,8} this corresponds to the energy of the first spinallowed band v_{1}, for d^{2,7} to the difference v_{2} – v_{1} in weak fields or v_{3} – v_{1} in stronger fields (cf. Fig. 6). For a reliable determination of Racah B, the term interaction between T_{1g}(F) and T_{1g}(P) has to be taken into account. Thus it can be obtained by solving the respective equations (e.g. Lever 1968, 1984) or by “fitting” the energy levels to the respective TanabeSugano diagram. Examples are given, e.g., by Lever (1968, 1984), Gade (1998) or Figgis & Hitchman (2000). In practice, leastsquares fitting of Dq and B (and of distortion parameters for lower symmetries) using an appropriate computer program is a good choice (e.g. Wildner, 1996b). If possible, Racah C can be subsequently estimated from properly assigned spinforbidden transitions. In case of d^{5} systems with the ^{6}A_{1g}(^{6}S) ground state no spinallowed band is observed and hence the correct assignment of the spinforbidden bands is essential even for extraction of Dq and Racah B. The latter parameter can be approximated from the energy difference (v_{5} – v_{3}) between the field independent levels ^{4}E_{g} + ^{4}A_{1g}(^{4}G) (v_{3}) and ^{4}E_{g}(^{4}D) (v_{5}), which equals 7B. Subsequently, Dq may be calculated by solving the algebraic expressions for the ^{4}T_{1g} (v_{1}) and ^{4}T_{2g} (v_{2}) levels, both originating from the ^{4}G freeion term. Anyway, the leastsquares fitting of Dq, B and C is the method of choice for d^{5} cases. For further details refer to the textbooks cited above.
As discussed above, the magnitude of the CF splitting parameter Δ_{i} (10Dq) depends on various factors, e.g. the type of coordination polyhedron and the M–L bond lengths. Similarly, the particular types of the central transition metal ion and of the coordinating ligand atoms play also an important role. Experimental data (e.g. Lever, 1984) indicate that for a given ligand, Dq increases approximately as follows: (i) about 30–80% with the oxidation state of a transition cation (M^{2+} < M^{3+} < M^{4+}), (ii) up to 100% with a change from the highspin to lowspin configuration (M_{hs} << M_{ls}), and (iii) about 30–60% on going from the 3d^{N} series to higher d element series (M^{3}^{d}^{} < M^{4}^{d}^{} < M^{5}^{d}). In the first two cases the concurrent reduction in the M–L bond lengths also contributes to an increase in Dq. Within a given transition series no systematic variation of Dq for a given oxidation state is observed. The “spectrochemical series of central atoms” given below summarises the observed variation of Dq with the transition ion, i.e.,
A similar variation with the ligand type can be observed and is summarised in the “spectrochemical series of ligands”. Jørgensen (1971) and others established such series involving many anionic groups and molecular ligands, so not directly applicable to geoscientific and related systems. The following sequence is found for the geoscientifically relevant ligands:
The oxygenbased ligands O^{2–}, OH^{−}, and H_{2}O produce crystal fields of comparable strength and hence their ordering is not clearcut, especially the sequence of O^{2–} and OH^{−}, and may vary in each individual case. The above two spectrochemical series have also been quantitatively factorised (J⊘rgensen, 1971), allowing a rough estimation of 10Dq for various metalligand combinations. The relative influence of the water molecule on the CF strength is usually set to f = 1.0, giving f = 0.8 for Cl^{−} and f = 0.9 for F^{−}. Note that the whole series of ligands ranges from f = 0.72 for Br^{−} up to f = 1.70 for CN^{−}. This fact clearly demonstrates that in minerals and related compounds the weak to intermediate crystal fields prevail. Predictions obtained from applications of the two series should, however, be treated with caution in view of their approximate character.
In the materials relevant for geosciences, the perturbations due to the d–d interelectron repulsion are of primary importance. The resulting energy splittings of free transition ions into spectroscopic terms can be described by the Racah parameters B and C. Parameters extracted from the atomic spectra of free gaseous transition ions are denoted B_{0} and C_{0} (or B_{fi} and C_{fi}). For the description of the spinallowed terms with maximum spin multiplicity, only B is needed and hence C is mostly given as a multiple of B, C ≈ 4B, and not separately discussed. For the free ions, the Racah parameters increase with the oxidation state, with the atomic number Z, and consequently also with the transition series. However, upon insertion of a transition metal ion into a crystalline environment, the d–d interelectronic repulsion is significantly modified by covalent contributions to the M–L bonding character in comparison with the perfectly ionic case of a free ion. The M–L bonding in a crystal field with increasingly covalent character can be interpreted in terms of a radial expansion of the d electron density, which consequently reduces the interelectronic repulsion effects. This is called the “nephelauxetic effect”, referring to the Greek word for “cloud expansion”. Hence, the values of the effective Racah parameters in crystals are reduced as compared with their freeion values, i.e., B_{cryst} < B_{0}, and the ratio β = B_{cryst}/B_{0} is used as sensitive indicator of covalency affecting a transition metal complex. However, as Reid & Newman (2000) pointed out, assuming that covalency makes an important positive contribution to Dq (cf. Newman & Ng, 2000b), then the interpretation of a covalencydependent expansion of the d electron cloud is in contradiction to the sequence of the magnitude of covalency as expressed by the spectrochemical series of ligands. Reid & Newman (2000) then argue that the reductions of interelectronic repulsion are mainly due to screening by the ligand charge clouds, i.e., the reductions are correlated to ligand polarisabilities.
Similarly to the spectrochemical series, the “nephelauxetic series of ligands” and the “nephelauxetic series of central atoms” have been established. In agreement with the arguments just cited above, the sequence of increasing nephelauxetic effect is related to the polarisability of the ligands, and for geoscientifically relevant ligands the ratio β decreases as follows:
Likewise, for a given ligand the central 3d^{N} cations can be arranged with decreasing value of β, i.e.,
Hence, contrary to the situation for free ions, B_{cryst} is strongly reduced with the increasing polarising power of the central cation.
It has to be mentioned that the description of the interelectronic repulsion effects by only two parameters is an (over)simplification. Actually, ten interelectronic repulsion parameters would be necessary to reflect the d–d electron interactions in O_{h} symmetry properly, even more parameters for lower symmetries (Griffith, 1961; Lever, 1984). Since the number of transitions observed in an optical absorption spectrum is usually much too small to extract so many interelectron and CF parameters, this simplification is a necessity. However, for the spinflip transitions occurring only within the t_{2g} or e_{g} orbital subset, an additional Racah B parameter and hence also β is applicable and can be extracted from highquality spectral data. This additional parameter reflects the phenomenon described in crystal field theory as “differential radial expansion”, i.e., a different bonding character and orbital expansion for the t_{2g} orbitals as compared with that for the e_{g} orbitals (J⊘rgensen, 1971; Lever, 1984). It is labelled B_{55} (and β_{55}) for the spinflips in the t_{2g} subset (e.g. in d^{3} systems) and B_{33} (and β_{33}) for those in the e_{g} subset (e.g. in d^{8} systems). To distinguish the “usual” Racah B parameter (and β) relevant for the spinallowed and other spinforbidden transitions, it is then labelled B_{35} (and β_{35}). Following the reasoning by Reid & Newman (2000) above, the effect of “differential radial expansion” has to be interpreted rather as the consequence of the anisotropic nature of ligand polarisation. These authors refer to such anisotropic twoelectron interactions within the scope of the “correlation crystal field” concept.
The effect of the SO coupling (ξ) strongly increases with the atomic number Z and with the oxidation state within a transition series, as well as between subsequent d^{N}^{} series. For the free ions, ξ ranges from 155 cm^{−1} to 830 cm^{−1} in the 3d^{N} series, around 500–2000 cm^{−1} for 4d^{N} and 1500–5000 cm^{−1} for 5d^{N} elements. Similar to the Racah parameters, the effective value ξ_{cryst} for ions in a CF is reduced from the free ion value (ξ_{0}), but reliable estimates of ξ_{cryst} are rare.
Dq_{oct} (O^{2})  B_{0} [1]  C_{0}/B_{0}[1]  B_{(35)}(O^{2})  B_{55,33}(O^{2})  C(O^{2})  ξ_{0}[1]  

Ti^{3+}  17002050            155 
V^{3+}  17001850  886  4.0  550700    25002850  210 
Cr^{3+}  15501850  933  4.0  450700  710760  28003170  275 
Mn^{4+}  18502200  1088  4.1  400600  790870  23503000  415 
Cr^{2+}  ∼ 1000  796  4.2        230 
Mn^{3+}  17002100  950  4.3  ∼ 800    ∼ 2600  355 
Mn^{2+}  650850  859  4.1  570790    32003770  300 
Fe^{3+}  12501450  1029  4.1  550650    ∼ 2900  460 
Fe^{2+}  8001040  897  4.3        400 
Co^{2+}  700920  989  4.3  750900    32503800  515 
Ni^{2+}  750900  1042  4.4  800950  750900  33503800  630 
Cu^{2+}  12001300  1240  3.8        830 
Dq_{oct} (O^{2})  B_{0} [1]  C_{0}/B_{0}[1]  B_{(35)}(O^{2})  B_{55,33}(O^{2})  C(O^{2})  ξ_{0}[1]  

Ti^{3+}  17002050            155 
V^{3+}  17001850  886  4.0  550700    25002850  210 
Cr^{3+}  15501850  933  4.0  450700  710760  28003170  275 
Mn^{4+}  18502200  1088  4.1  400600  790870  23503000  415 
Cr^{2+}  ∼ 1000  796  4.2        230 
Mn^{3+}  17002100  950  4.3  ∼ 800    ∼ 2600  355 
Mn^{2+}  650850  859  4.1  570790    32003770  300 
Fe^{3+}  12501450  1029  4.1  550650    ∼ 2900  460 
Fe^{2+}  8001040  897  4.3        400 
Co^{2+}  700920  989  4.3  750900    32503800  515 
Ni^{2+}  750900  1042  4.4  800950  750900  33503800  630 
Cu^{2+}  12001300  1240  3.8        830 
[1] Figgis & Hitchman (2000)
Table 12 summarises the available data on the freeion parameters B_{0}, C_{0}/B_{0} (Racah) and ξ_{0} (SO), as well as Dq (CF) and the effective Racah parameters in octahedral oxygenbased ligand fields for several important 3d^{N} cations.
References
The authors thank U. Hålenius, Stockholm, and D.J. Newman, Southampton, for useful comments which helped us to improve the manuscript. MW and MA gratefully acknowledge financial support to MA by a research fellowship from the Austrian Science Fund (FWF) for the project “Superposition model analysis for application in mineralogy”, no. P13976CHE.
Figures & Tables
wavenumber [cm^{−1}]  wavelength λ [nm]  frequency v [10^{14} Hz]  energy E [10^{19} J]  energy E [eV] 

40000  250  11.99  7.95  4.96 
25000  400  7.49  4.97  3.10 
18182  550  5.45  3.61  2.25 
14286  700  4.28  2.84  1.77 
10000  1000  3.00  1.99  1.24 
4000  2500  1.20  0.79  0.50 
wavenumber [cm^{−1}]  wavelength λ [nm]  frequency v [10^{14} Hz]  energy E [10^{19} J]  energy E [eV] 

40000  250  11.99  7.95  4.96 
25000  400  7.49  4.97  3.10 
18182  550  5.45  3.61  2.25 
14286  700  4.28  2.84  1.77 
10000  1000  3.00  1.99  1.24 
4000  2500  1.20  0.79  0.50 
Z  element  M^{0}  M^{+}  M^{2+}  M^{3+}  M^{4+}  M^{5+}  M^{6+}  M^{7+} 

21  Sc  [Ar] 3d^{1}4s^{2}  [Ar]  
22  Ti  [Ar] 3d^{2}4s^{2}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
23  V  [Ar] 3d^{3}4s^{2}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
24  Cr  [Ar] 3d^{5}4s^{1}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar]  
25  Mn  [Ar] 3d^{5}4s^{2}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{1}  [Ar]  
26  Fe  [Ar] 3d^{6}4s^{2}  [Ar] 3d^{6}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{2}  
27  Co  [Ar] 3d^{7}4s^{2}  [Ar] 3d^{7}  [Ar] 3d^{6}  
28  Ni  [Ar] 3d^{8}4s^{2}  [Ar] 3d^{9}  [Ar] 3d^{8}  [Ar] 3d^{7}  [Ar] 3d^{6}  
29  Cu  [Ar] 3d^{10}4s^{1}  [Ar] 3d^{10}  [Ar] 3d^{9}  [Ar] 3d^{8}  
30  Zn  [Ar] 3d^{10}4s^{2}  [Ar] 3d^{10} 
Z  element  M^{0}  M^{+}  M^{2+}  M^{3+}  M^{4+}  M^{5+}  M^{6+}  M^{7+} 

21  Sc  [Ar] 3d^{1}4s^{2}  [Ar]  
22  Ti  [Ar] 3d^{2}4s^{2}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
23  V  [Ar] 3d^{3}4s^{2}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar] 3d^{1}  [Ar]  
24  Cr  [Ar] 3d^{5}4s^{1}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{2}  [Ar]  
25  Mn  [Ar] 3d^{5}4s^{2}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{3}  [Ar] 3d^{1}  [Ar]  
26  Fe  [Ar] 3d^{6}4s^{2}  [Ar] 3d^{6}  [Ar] 3d^{5}  [Ar] 3d^{4}  [Ar] 3d^{2}  
27  Co  [Ar] 3d^{7}4s^{2}  [Ar] 3d^{7}  [Ar] 3d^{6}  
28  Ni  [Ar] 3d^{8}4s^{2}  [Ar] 3d^{9}  [Ar] 3d^{8}  [Ar] 3d^{7}  [Ar] 3d^{6}  
29  Cu  [Ar] 3d^{10}4s^{1}  [Ar] 3d^{10}  [Ar] 3d^{9}  [Ar] 3d^{8}  
30  Zn  [Ar] 3d^{10}4s^{2}  [Ar] 3d^{10} 
m_{l}  

d^{N}  2  1  0  1  2  L = Σm_{l}  2L + 1  S = Σm_{s}  2S + 1  ground term (degeneracy) 
d^{1}  ↑  2  5  1/2  2  ^{2}D (10)  
d^{2}  ↑  ↑  3  7  1  3  ^{3}F (21)  
d^{3}  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28)  
d^{4}  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25)  
d^{5}  ↑  ↑  ↑  ↑  ↑  0  1  5/2  6  ^{6}S (6) 
d^{6}  ↓↑  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25) 
d^{7}  ↓↑  ↓↑  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28) 
d^{8}  ↓↑  ↓↑  ↓↑  ↑  ↑  3  7  1  3  ^{3}F (21) 
d^{9}  ↓↑  ↓↑  ↓↑  ↓↑  ↑  2  5  1/2  2  ^{2}D (10) 
m_{l}  

d^{N}  2  1  0  1  2  L = Σm_{l}  2L + 1  S = Σm_{s}  2S + 1  ground term (degeneracy) 
d^{1}  ↑  2  5  1/2  2  ^{2}D (10)  
d^{2}  ↑  ↑  3  7  1  3  ^{3}F (21)  
d^{3}  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28)  
d^{4}  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25)  
d^{5}  ↑  ↑  ↑  ↑  ↑  0  1  5/2  6  ^{6}S (6) 
d^{6}  ↓↑  ↑  ↑  ↑  ↑  2  5  2  5  ^{5}D (25) 
d^{7}  ↓↑  ↓↑  ↑  ↑  ↑  3  7  3/2  4  ^{4}F (28) 
d^{8}  ↓↑  ↓↑  ↓↑  ↑  ↑  3  7  1  3  ^{3}F (21) 
d^{9}  ↓↑  ↓↑  ↓↑  ↓↑  ↑  2  5  1/2  2  ^{2}D (10) 
d^{N}, d^{10}^{N}  Freeion spectroscopic terms ^{2}^{S}^{+1}L  deg_{tot} 

d^{1}, d^{9}  ^{2}D  10 
d^{2}, d^{8}  ^{3}F, ^{3}P, ^{1}G, ^{1}D, ^{1}S  45 
d^{3}, d^{7}  ^{4}F, ^{4}P, ^{2}H, ^{2}G, ^{2}F, ^{2}D_{a}, ^{2}D_{b}, ^{2}P  120 
d^{4}, d^{6}  ^{5}D, ^{3}H, ^{3}G, ^{3}F_{a}, ^{3}F_{b}, ^{3}D, ^{3}P_{a}, ^{3}P_{b}, ^{1}I, ^{1}G_{a}, ^{1}G_{b}, ^{1}F, ^{1}D_{a}, ^{1}D_{b}, ^{1}S_{a}, ^{1}S_{b}  210 
d^{5}  ^{6}S, ^{4}G, ^{4}F, ^{4}D, ^{4}P, ^{2}I, ^{2}H, ^{2}G_{a}, ^{2}G_{b}, ^{2}F_{a}, ^{2}F_{b}, ^{2}D_{a}, ^{2}D_{b}, ^{2}D_{c}, ^{2}P, ^{2}S  252 
d^{N}, d^{10}^{N}  Freeion spectroscopic terms ^{2}^{S}^{+1}L  deg_{tot} 

d^{1}, d^{9}  ^{2}D  10 
d^{2}, d^{8}  ^{3}F, ^{3}P, ^{1}G, ^{1}D, ^{1}S  45 
d^{3}, d^{7}  ^{4}F, ^{4}P, ^{2}H, ^{2}G, ^{2}F, ^{2}D_{a}, ^{2}D_{b}, ^{2}P  120 
d^{4}, d^{6}  ^{5}D, ^{3}H, ^{3}G, ^{3}F_{a}, ^{3}F_{b}, ^{3}D, ^{3}P_{a}, ^{3}P_{b}, ^{1}I, ^{1}G_{a}, ^{1}G_{b}, ^{1}F, ^{1}D_{a}, ^{1}D_{b}, ^{1}S_{a}, ^{1}S_{b}  210 
d^{5}  ^{6}S, ^{4}G, ^{4}F, ^{4}D, ^{4}P, ^{2}I, ^{2}H, ^{2}G_{a}, ^{2}G_{b}, ^{2}F_{a}, ^{2}F_{b}, ^{2}D_{a}, ^{2}D_{b}, ^{2}D_{c}, ^{2}P, ^{2}S  252 
Term (L)  CF states 

S  A_{1g} 
P  T_{1g} 
D  E_{g}, T_{2g} 
F  A_{2g}, T_{1g}, T_{2g} 
G  A_{1g}, E_{g}, T_{1g}, T_{2g} 
H  E_{g}, T_{1g}, T_{1g}, T_{2g} 
I  A_{1g}, A_{2g}, E_{g}, T_{1g}, T_{2g}, T_{2g} 
Term (L)  CF states 

S  A_{1g} 
P  T_{1g} 
D  E_{g}, T_{2g} 
F  A_{2g}, T_{1g}, T_{2g} 
G  A_{1g}, E_{g}, T_{1g}, T_{2g} 
H  E_{g}, T_{1g}, T_{1g}, T_{2g} 
I  A_{1g}, A_{2g}, E_{g}, T_{1g}, T_{2g}, T_{2g} 
O_{h}  D_{3}_{d}  C_{3}_{v}  C_{3} 

A_{1g}  A_{1g}  A_{1}  A 
A_{2g}  A_{2g}  A_{2}  A 
E_{g}  E_{g}  E  E 
T_{1g}  A_{2g} +E_{g}  A_{2} + E  A + E 
T_{2g}  A_{1g} + E_{g}  A_{1} + E  A + E 
O_{h}  D_{3}_{d}  C_{3}_{v}  C_{3} 

A_{1g}  A_{1g}  A_{1}  A 
A_{2g}  A_{2g}  A_{2}  A 
E_{g}  E_{g}  E  E 
T_{1g}  A_{2g} +E_{g}  A_{2} + E  A + E 
T_{2g}  A_{1g} + E_{g}  A_{1} + E  A + E 
C_{3}  E  C_{3}  Vector components (e.g. translations, dipole moment operator M, p orbitals and rotations R_{i})  Tensor components (e.g. d orbitals)  

A  1  1  1  z, R_{z}  x^{2}+y^{2}, z^{2} 
E  (x^{2}y^{2}, xy) (yz, xz) 
C_{3}  E  C_{3}  Vector components (e.g. translations, dipole moment operator M, p orbitals and rotations R_{i})  Tensor components (e.g. d orbitals)  

A  1  1  1  z, R_{z}  x^{2}+y^{2}, z^{2} 
E  (x^{2}y^{2}, xy) (yz, xz) 
Dq_{oct} (O^{2})  B_{0} [1]  C_{0}/B_{0}[1]  B_{(35)}(O^{2})  B_{55,33}(O^{2})  C(O^{2})  ξ_{0}[1]  

Ti^{3+}  17002050            155 
V^{3+}  17001850  886  4.0  550700    25002850  210 
Cr^{3+}  15501850  933  4.0  450700  710760  28003170  275 
Mn^{4+}  18502200  1088  4.1  400600  790870  23503000  415 
Cr^{2+}  ∼ 1000  796  4.2        230 
Mn^{3+}  17002100  950  4.3  ∼ 800    ∼ 2600  355 
Mn^{2+}  650850  859  4.1  570790    32003770  300 
Fe^{3+}  12501450  1029  4.1  550650    ∼ 2900  460 
Fe^{2+}  8001040  897  4.3        400 
Co^{2+}  700920  989  4.3  750900    32503800  515 
Ni^{2+}  750900  1042  4.4  800950  750900  33503800  630 
Cu^{2+}  12001300  1240  3.8        830 
Dq_{oct} (O^{2})  B_{0} [1]  C_{0}/B_{0}[1]  B_{(35)}(O^{2})  B_{55,33}(O^{2})  C(O^{2})  ξ_{0}[1]  

Ti^{3+}  17002050            155 
V^{3+}  17001850  886  4.0  550700    25002850  210 
Cr^{3+}  15501850  933  4.0  450700  710760  28003170  275 
Mn^{4+}  18502200  1088  4.1  400600  790870  23503000  415 
Cr^{2+}  ∼ 1000  796  4.2        230 
Mn^{3+}  17002100  950  4.3  ∼ 800    ∼ 2600  355 
Mn^{2+}  650850  859  4.1  570790    32003770  300 
Fe^{3+}  12501450  1029  4.1  550650    ∼ 2900  460 
Fe^{2+}  8001040  897  4.3        400 
Co^{2+}  700920  989  4.3  750900    32503800  515 
Ni^{2+}  750900  1042  4.4  800950  750900  33503800  630 
Cu^{2+}  12001300  1240  3.8        830 
[1] Figgis & Hitchman (2000)
Contents
Spectroscopic methods in mineralogy
Spectroscopic methods provide information about the local structure of minerals. The methods do not depend on longrange periodicity or crystallinity. The geometric arrangement of atoms in a mineral phase is only one aspect of its constitution. Its vibrational characteristic, electronic structure and magnetic properties are of greatest importance when we consider the behaviour of minerals in dynamic processes. The characterisation of the structural and physicochemical properties of a mineral requires the application of several complementary spectroscopic techniques. However, it is one of the main aims of this School to demonstrate that different spectroscopic methods work on the same basic principles. Spectroscopic techniques represent an extremely rapidly evolving area of mineralogy and many recent research efforts are similar to those in materials science, solid state physics and chemistry. Applications to different materials of geoscientific relevance have expanded by the development of microspectroscopic techniques and by in situ measurements at low to hightemperature and highpressure conditions.