Theory crystal field

Crystal field theory was introduced in the late 1920s by Bethe and Van Vleck and, although initially formulated and applied by physicists, incorporates the inorganic paradigm of primary concern with the consequences at the metal. The crystal field theory 

Each electron placed in one of the t2g orbitals is stabilised by a total of- A, whereas those placed in the higher energy eg orbitals are each destabilised by a total of A. 

We use the descriptors t2 and e for the two groups of orbitals. The subscript g is not used for systems which do not possess an inversion centre. The splitting of the orbitals is thus exactly the reverse of that observed in an octahedral ligand field. Each electron placed in 

In the early use of VB theory, complexes in which the electronic configuration of the metal ion was the same as that of the free gaseous atom were called ionic complexes, while those in which the electrons had been paired up as far as possible were called covalent complexes. Later, first row metal complexes in which ligand electrons entered 3 i orbitals (as in [Fe(CN)g] ) were termed inner orbital complexes, and those in which Ad orbitals were occupied 

With the electrons from the ligands included and a hybridization scheme applied for an octahedral complex, the diagram becomes  

This diagram is appropriate for all octahedral Cr(III) complexes because the three 3d electrons always singly occupy different orbitals. 

For octahedral Fe(III) complexes, we must account for the existence of both high- and low-spin complexes. The electronic configuration of the free Fe ion is  

For a low-spin octahedral complex such as [Fe(CN)g], we can represent the electronic configuration by means of the following diagram where the electrons shown in red are donated by the ligands  

Crystal field theory assumes that all M-L interactions are purely electrostatic in nature. More specifically, it considers the electrostatic effect of a field of ligands on the energies of a metals valence-shell orbitals. To discuss CFT, we need only be aware of two fundamental concepts (1) the coulombic theory of electrostatic interactions and (2) the shapes of the valence orbitals of transition metals—that is, the nd orbitals ( = 3 for the first row of transition metals, etc.). The first concept involves only the familiar ideas of the repulsion of like and the attraction of dislike electrical charges. Quantitatively, Coulomb s law states that the potential energy of two charges Qj and Q2 separated by a distance r is given by the formula shown in Equation (4.3)  

The second concept, the shapes of d orbitals, requires a little more development. 

When the d orbitals of a metal ion are placed in an octahedral field of ligand electron pairs, any electrons in these orbitals are repelled by the field. As a result, the d -y and d orbitals, which have symmetry, are directed at the surrounding ligands and are raised 

For better comprehension of the crystal field theory, we must reconsider to some extent the wavelike behavior of electrons. 

The wave function of the hydrogen atom in the ground state (n = 1) is given by 

In equation 1.116, Uq is Bohr s radius for the hydrogen ion in the ground state 

We have already seen (section 1.1.2) that the wave equation (eq. 1.2, 1.4) is function of both time and spatial coordinates. However, the wave function may be rewritten in the form 

The product in equation 1.121, which is independent of time, represents the density of distribution—i.e., the probability of finding a given particle in the region specified by the spatial coordinates of ip The application of wave equation 1.10 to hypothetical atoms 

A second approach to the bonding in complexes of the d-block metals is crystal field theory. This is an electrostatic model and simply uses the ligand electrons to create an 

Crystal field theory is an electrostatic model which predicts that the d orbitals in a metal complex are not degenerate. The pattern of splitting of the d orbitals depends on the crystal field, this being determined by the arrangement and type of ligands. 

If the electrostatic field created by die point charge ligands is spherical. 

CHEMICAL AND THEORETICAL BACKGROUND Box 21.1 A reminder about symmetry labels 

The subscript g means gerade and the subscript u means ungerade. Gerade and ungerade designate the behaviour of the wavefxmction under the operation of inversion, and denote the parity (even or odd) of an orbital. 

When the ligands interact more strongly the MOs of the ligands must be taken into account. This type of MO theory is referred to as ligand field theory. 

Many transition-metal complexes exhibit paramagnetism, as described in Sections 9.8 and 23.1. In such compounds the metal ions possess some number of unpaired electrons. It is possible to experimentally determine the number of unpaired electrons per metal ion from the measured degree of paramagnetism, and experiments reveal some interesting comparisons. 

Compounds of the complex ion [Co(CN)g] have no unpaired electrons, for example, but compounds of the [CoFg] ion have four unpaired electrons per metal ion. Both complexes contain Co(III) with a 3d electron configuration. — (Section 7.4) Clearly, there is a major difference in the ways in which the electrons are arranged in these two cases. Any successful bonding theory must explain this difference, and we present such a theory in the next section. 

How would this absorbance spectrum change if you decreased the concentration of the Ui(H20)e] in 

yellow absorbed violet and red light travel to eye, solution appears red-violet 

Scientists have long recognized that many of the magnetic properties and colors of transition-metal complexes are related to the presence of d electrons in the metal cation. In this section we consider a model for bonding in transition-metal complexes, crystal-field theory, that accounts for many of the observed properties of these substances. Because the predictions of crystal-field theory are essentially the same as those obtained with more advanced molecular-orbital theories, crystal-field theory is an excellent place to start in considering the electronic structure of coordination compounds. 

Calculations of coordinate bond energies can be made using classical potential energy equations that take into account the attractive and repulsive interactions between charged particles (10)  

This refinement of electrostatic theory was first recognized and used by the physicists Bethe and Van Vleck in 1930 to explain colors and magnetic properties of crystalline solids. Their theory known as crystal field theory was proposed at about the same time as-or even a little earlier than-VBT, but it took about twenty years for the CFT to be recognized and used by chemists. Perhaps this was because CFT was written for physicists and VBT gave such a satisfying pictorial representation of the bonded atoms. 

In 1951, several theoretical chemists working independently used CFT to interpret spectra of transition-metal complexes with such success that there followed an immediate avalanche of research activity in the area. It soon 

These eg orbitals point directly at the F ligands, whereas the dxy dxz, and dyz 

Sponge ball under presisure from a spherical shell 

The interaction of a free metal ion in the gas phase with a sphere of negative charge causes the energy of the d-orbitals to increase as a result of the smaller value of r. Redistribution of the negative charge in an octahedral CF causes some of the orbitals to be raised with respect to the barycenter, whiie others are stabilized. The splitting between the two energy levels is defined as or lODq. 

The relative energies of the five d-orbitals (a) in a weak CF, as is the case for the high-spin [FeFg] complex ion and (b) in a strong CF, as is the case for the low-spin [Fe(CN)g] ion. The dashed line represents the barycenter. 

Weak-field high-spin Act 0 Strong-field low-spin A) Aoi 

The Pg term is related to the loss of the exchange energy when electrons having parallel spins are forced to be antiparallel. Statistically, the exchange energy is related 

TABLE 16.2 Pairing energies (kJ/mol) for selected metal Ions in the gas phase.  

3 The lattice energies of bivalent first transition series metal chlorides, MCI2, where is 

CaCb ScCb TiCb VCI2 CrCl2 MnCl2 FeCb C0CI2 NiCb CuCb ZnCb 

Estimate the crystal field strengths of the chloride ions on and Co.  

4 A spinel is a metal oxide, M3O4, which has a nearly close-packed array of oxide ions. The three metal ions occupy one tetrahedral and two octahedral sites. The formula may then be written as A[BC]04, where the ions in octahedral sites are enclosed in brackets. Furthermore, a spinel in which the octahedral sites are occupied by M (III) ions is known 

For Mn2Fe04, calculate the total crystal field stabilization energy (CFSE) in units of Ao (or Dq) for all permutations of ions and oxidation states. (As indicated earlier, only M(II) and M(III) states need to be considered.) Other assumptions are At = Ao5 (Ao)pe = (Ao)Mn5 (Ao)2-i- = (Ao)3+ and both iron and manganese oxides are high-spin cases. Deduce the correct structure on the basis of the calculated CFSE. Is the deduced spinel normal or inverse  

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For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

For transition metal complexes, techniques derived from a crystal-field theory or ligand-field theory description of the molecules have been created. These tend to be more often qualitative than quantitative. 

Transition metals readily form complexes, such as [Fe(CN)6], the ferrocyanide ion, Ni(CO)4, nickel tetracarbonyl, and [CuC ], the copper tetrachloride ion. MO theory applied to such species has tended to be developed independently. It is for this reason that the terms crystal field theory and ligand field theory have arisen which tend to disguise the fact that they are both aspects of MO theory. 

Color from Transition-Metal Compounds and Impurities. The energy levels of the excited states of the unpaked electrons of transition-metal ions in crystals are controlled by the field of the surrounding cations or cationic groups. Erom a purely ionic point of view, this is explained by the electrostatic interactions of crystal field theory ligand field theory is a more advanced approach also incorporating molecular orbital concepts. 

E. Basolo and R. G. Pearson, Mechanisms of Inorganic Reactions, 2nd ed., John Wiley Sons, Inc., New York, 1967. An excellent volume that stresses the reactions of complexes ia solution a background and a detailed theory section is iacluded that is largely crystal field theory, but some advantages and disadvantages of molecular orbital theory are iacluded. 

Herzfeld, C. M., and Meijer, P. H. E., Group Theory and Crystal Field Theory, in F. Seitz and D. Turnbull, eds., Solid State Physics, Yol. 12, Academic Press, Mew York, 1961. 

Crystal field splitting parameter, 2, 309 Crystal field theory, 1, 215-221 angular overlap model, 1, 228 calculations, 1, 220 generality, 1,219 low symmetry, 1,220 /-orbital, 1, 231 Crystal hydrates, 2, 305,306 bond distances, 2, 307 Crystals... 

There are two major theories of bonding in d-metal complexes. Crystal field theory was first devised to explain the colors of solids, particularly ruby, which owes its color to Cr3+ ions, and then adapted to individual complexes. Crystal field theory is simple to apply and enables us to make useful predictions with very little labor. However, it does not account for all the properties of complexes. A more sophisticated approach, ligand field theory (Section 16.12), is based on molecular orbital theory. 

FIGURE 16.24 In the crystal field theory of complexes, the lone pairs of electrons that serve as the Lewis base sites on the ligands (a) are treated as equivalent to point negative charges (b). 

The effects of the bonding electrons upon the d electrons is addressed within the subjects we call crystal-field theory (CFT) or ligand-field theory (LFT). They are concerned with the J-electron properties that we observe in spectral and magnetic measurements. This subject will keep us busy for some while. We shall return to the effects of the d electrons on bonding much later, in Chapter 7. 

Crystal-field theory (CFT) was constructed as the first theoretical model to account for these spectral differences. Its central idea is simple in the extreme. In free atoms and ions, all electrons, but for our interests particularly the outer or non-core electrons, are subject to three main energetic constraints a) they possess kinetic energy, b) they are attracted to the nucleus and c) they repel one another. (We shall put that a little more exactly, and symbolically, later). Within the environment of other ions, as for example within the lattice of a crystal, those electrons are expected to be subject also to one further constraint. Namely, they will be affected by the non-spherical electric field established by the surrounding ions. That electric field was called the crystalline field , but we now simply call it the crystal field . Since we are almost exclusively concerned with the spectral and other properties of positively charged transition-metal ions surrounded by anions of the lattice, the effect of the crystal field is to repel the electrons. 

Except to note that the occurrence of the coefficients 15 and 10 in 155 and lOD obviate the need for fractions here or elsewhere in crystal-field theory thus, they are there for reasons of convenience and definition only. 

Much useful understanding of the processes of crystal-field theory, however, can be had from a study of just the free-ion ground terms. Application of the simple process in Section 3.7 identifies the ground terms for d free ions as D, D,... 

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