LIGAND FIELD THEORY

Structure, Bonding, and Properties

Rule 4: Linking of polyhedra having different cations

1.6. LIGAND FIELD THEORY

Ligand field theory is the basis for describing structure evaluation in functional materials. The concept of ligand field theory is equivalent to that of the valence shell electron-pair repulsion theory, and it describes how the d shell electrons are distributed to attain a minimum repulsion with each other and with the bonding electron pairs.

(a) Linear arrangement (2) (b) Triangle (3) (c) Tetrahedron (4)

(d) Trigonal bipyramid (5) (e) Square pyramid (5) (f) Octahedron (6)

(g) Square antiprism (8) (h) Triply capped trigonal prism _(i) Icosahedron (12) Figure 1.8. Possible arrangements of points on the surface of a sphere with minimum repulsion energy_

31

STRUCTURE, BONDING, AND PROPERTIES

32

CHAPTER 1

The relative orientation of the regions with high charge density of d electrons and of bonding electrons around an atom can be described with the aid of a coordination system having its origin in the center of the atom. Two sets of d orbitals generally can be distinguished: the first set consists of two orbitals oriented along the coordinated axes, and the second set consists of three orbitals oriented toward the centers of the edges of a circumscribed cube (Fig. 1.9).

1.6.1. OCTAHEDRAL COORDINATION

If an atom has six ligands, the mutual repulsion of the six bonding electron pairs results in an octahedral coordinated polyhedron. The positions of the ligands are to the points on the axes of the coordinated system. If nonbonding electrons exist, they will prefer the orbitals dxy, dyz , d_xz because the regions of high charge density of the other two d orbitals are close to the bonding electron pairs. The three orbitals favored energetically are called t2g orbitals (this is a symbol for orbital symmetry, and the t specifies a triply specifies state); the other two are e_g orbitals (e means doubly degenerate).

The energy difference between the occupation of a t_2g and an eg orbital is defined as Llo (Fig. 1.10). The value of Llo depends on the repulsion exercised by the bonding electron pairs on the d electrons. In comparison to a transition metal element the bonded ligand atoms are usually much more electronegative. The centers of the charge of the bonding electron pairs are much closer to ligand atoms, especially when they are strongly electronegative. Therefore, a decreasing influence on the d electrons of transition metal

Figure 1.9. Contour maps for the distributions of electron density for 3d orbitals (after Brickmann et al., 1978).

E dz2 dx2_y2

e

^g

1:

~o

dxy dxz dyz

t 2g Figure 1.10. Energy levels for the orbitals of 3d electrons.

cation at the center results in a decrease of ~o with increasing ligand electronegativity.

Decreasing ~o values also result in increasing sizes of the ligand atoms; in this case the electron pairs are distributed over a larger space, the difference of their repulsive action on a t2g and an eg orbital is less. In the presence of multiple bonds between the metal atom and the ligands, the electron density of the bonds is especially high and their action is large. Since ~o is a value that can be measured directly using spectroscopic methods, the activities of different kinds of ligands are obtained. For instance, by photoexcitation of an electron from the t2g to the eg level, ~o is calculated by hv. The spectrochemical series is obtained by ordering different ligands according to decreasing ~o:

Hund

s

rules are very useful for understanding the distribution of electrons in those orbitals.

Hund's first rule: out of the many states one can form by placing n electrons into the 2 (2L

+

1) levels of the partially filled shell, those that lie lowest in energy have the largest total spin S that is consistent with the Pauli exclusion principle.

Hund's second rule: The total orbital angular momentum Lt of the lowest-lying states has the largest value that is consistent with Hund 50 first rule and with the Pauli exclusion principle.

Pauli's exclusion principle: No more than one electron is permitted to occupy any single electron state defined by quantum numbers including spin.

When two or three nonbonding electrons are present, they will occupy two or three of the t2g orbitals, respectively, according to Hund's rules. This is more favorable than pairing electrons in one orbit because the pairing requires that the electrostatic repulsion between the two electrons be overcome. The energy necessary to include a second electron in an already occupied orbital is termed the electron pairing energy P. When four nonbonding electrons are present, there are two alternatives for the placement of the fourth electron. If P > ~o, it is an eg orbital and all four electrons will have parallel spins.

This is called a high-spin complex (Fig. l.l1a). If P < ~o, it is more favorable to form a low-spin complex in which there is no electron on the eg orbitals while two paired electrons are in the t2g orbitals (Fig. 1.11 b).

In a high-spin d⁴ complex only one of the two eg orbitals is occupied. If it is the dz²

orbital, it gives a strong repulsion on the bonding electrons of the two ligands on the z axis. These ligands are forced outwards, the coordination octahedron experiences an elongation along the z axis. This effect is known as the Jahn-Teller effect, which describes the distortion of the crystal to low the symmetry for removing the state degeneracy.

Instead of the dz2 orbital the dx2_1 orbital could have been occupied, which would have produced elongations along the x and y axes. However, a higher force is needed to stretch

33

STRUCTURE, BONDING, AND PROPERTIES

34

CHAPI'ER 1

(a) High-spin

(c)

+

+++

_{d4 high-spin}

[Cr(I1), Mn(III)]

(b) E

Llo

*** *+

_[Cu(I1)]

Low-spin

+++ +-

_{d7 low-spin}

[Ni(III)]

Figure 1.11. Electron distribution in (a) high-spin and (b) low spin molecular orbitals. (c) Examples of high- and low-spin electron distributions.

four bonds. Therefore stretching only two bonds is energetically favorable, and consequently only octahedra elongated in Qne direction are known so far.

The Jahn-Teller effect is always to be expected if degenerate orbitals are unevenly occupied with electrons. In fact it is observed for the electronic configurations shown in Fig. l.11c.

A Jahn-Teller distortion should also occur for configuration d1• The occupied orbital is a t2g orbital, for example dyz; this causes a repulsion on the ligands on the axes of y and z, which is only slightly larger than the force exerted along the x axis. The distorting force is usually not sufficient to introduce a perceptible effect. Ions like TiF63- or MoCl6 -,

show no detectable deviation from octahedral symmetry.

Neither the slightest Jahn-Teller distortion nor deviation from the ideal octahedral symmetry occurs if the t2g and eg orbitals are occupied evenly. This applies for the following electronic configurations: dO, d3 , d^S high-spin, d⁶ low-spin, d^8, and dlO. For configuration d^8, octahedral coordination rarely occurs. If there are different kinds of ligands, those which have the smaller influence according to the spectrochemical series prefer the positions with the stretched bonds.

1.6.2. TETRAHEDRAL COORDINATION

The four ligands of a tetrahedrally coordinated atom can be placed in four of the eight vertices of a cube. The orbitals dxy, dyz , ^{dxz (t2} orbitals), which are oriented toward the cube edges, are closer to the bonding electron pairs than the orbitals d_x2-y'2 and dz²

(e orbitals). Consequently, the t2 orbitals produce a larger repulsion and become energetically higher than the e orbitals. The consequence is opposite to that of octahedral coordination. The energy difference is termed Llt • Since none of the d orbitals is oriented toward a cube vertex, Llt < Llo is expected (for equal ligands, equal central atom and

equal bond lengths), or more specifically I1t ~ (4/9)110' Tetrahedral complexes are always high-spin complexes.

When the t2 orbitals are occupied unevenly, Jahn-Teller distortions occur. For the configuration d⁴ one of the t2 orbitals is unoccupied. For d⁹ one has single occupation and the rest double. As a consequence, the ligands have differing repulsion, and a flattened tetrahedron is formed (Fig. 1.12).

For the configuration d³ and d⁸ one t2 orbital has one electron more than the others.

An elongated tetrahedron is expected. The deformation is smaller than for d⁴ and d9,

because the deforming repulsion force is being exerted by only one electron instead of two. Due to the deformation force is small and the requirements of the packing in the crystal cause opposite deformations, so the observations could be different. For example, NiCIl- (d8) has been observed to have undistorted, slightly elongated or slightly flattened tetrahedra depending on the cation. For uneven occupation of the orbital distortions could also be expected, but the effect is even smaller and usually it is not detectable.

1.6.3. SQUARE COORDINATION

If the two ligands on the z axis of an octahedral complex are removed, the remaining ligands form a square. The repulsion between bonding electrons on the z axis ceases for the d_z2, the d_xz, and the dyz electrons. Only one orbital, namely d_xl_y2, still has a strong repulsion from the remaining bond electrons and is energetically unfavorable (Fig. 1.13).

Square coordination is the favored coordination for d⁸ configuration. This is probably the

Flattened tetrahedron

d

⁴

d

⁹

++- ##+

++

^Or

#*

Elongated tetrahedron

+ -+-+

^Or

^#+-+ #*

Figure 1.12. Iahn-Teller distortions of tetrahedral polyhedron. The arrowheads indicate the directions of displacements of the ligands due to repulsion by the nonbound d electrons. The spheres on the cube edges mark the centers of gravity of the charges of the t2 orbitals, a shadowed sphere being occupied by one more electron than an open sphere. (Reprinted with permission from Wiley & Sons, Inc.).

35

STRUCTURE, BONDING, AND PROPERTIES

36

CHAPTER I

E

Tetrahedron Octahedron Elongated octahedron Square

Figure 1.l3. Schematics of the relative energies of electrons in the d orbitals for different polyhedral geometries. The mean values of the energy levels for all term sequences are positioned on the dashed line.

(Reprinted with permission from Wiley & Sons, Inc.).

reason that NI(II), Pd(II), Pt(1I) and Au (II) complexes have square coordination rather than octahedral or tetrahedral complex.