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A Common Inorganic Chemistry Textbook Mistake: Incorrect Use of Pairing Energy in Crystal Field Stabilization Energy Expressions David Tudela Departamento de Química Inorgánica, Universidad Autónoma de Madrid, 28049-Madrid, Spain
Crystal field theory (CFT) is a purely electrostatic approach to the bonding in transition metal complexes. It considers the effect of the electric field created by the ligands (taken as negative point charges) on the energy of the metal ion d orbitals. Although it is a rough approximation, CFT gives satisfactory results, and because of its simplicity, it is easily understood by undergraduate students. Furthermore, CFT is a good introduction to more sophisticated bonding theories. When a transition metal ion is approached by a set of ligands, there is a large electrostatic attraction and the energy of the system is lowered, so that the complex is stable. Although the total energy of the system is lowered, the energy of the metal ion d orbitals is increased by the electrostatic repulsion between the negative charge on the ligands and the electrons in the metal’s d orbitals. If the field created by the ligands were spherically symmetrical, the d orbitals would remain degenerate, as they are in an isolated, gaseous metal ion. Nevertheless, if six ligands are at the corners of a regular octahedron, the five d orbitals are no longer degenerate, but they split into two sets of orbitals, the t 2g orbitals (dxz , dyz , and d xy) and the eg orbitals (dx2- y2 and dz2). The eg orbitals are exactly directed toward the ligands and they have, therefore, a higher energy than the t 2g orbitals, which are oriented between adjacent ligands. The energy difference between the two sets of orbitals is called ∆o (or 10Dq), and the energies of the t2g and eg levels are, respectively, 0.4∆o below and 0.6∆o above the energy of the unsplit d orbitals.
As a result of the splitting of the d orbitals, there is a crystal field stabilization energy (CFSE) relative to the hypothetical spherical field with no splitting. Each electron in a t 2g orbital lowers the energy of the system by 0.4∆ o, whereas each electron in an eg orbital raises the energy by 0.6∆o. In addition, each electron pair forced to be paired in the same orbital raises the energy of the system by the pairing energy P. For d4, d5, d6, and d7 configurations, if ∆o < P, the system is more stable if the electrons occupy the eg orbitals rather than being paired in the t2g orbitals, giving rise to high-spin complexes. If ∆o > P, a low-spin complex results in which the electrons are paired in the t 2g orbitals rather than occupying the higher-energy eg orbitals. CFSE values for octahedral complexes are collected in Table 1. Some of the most important inorganic chemistry textbooks (1–3) contain a mistake. They include, incorrectly, the pairing energy in expressions of the crystal field stabilization energy of octahedral d 6 (and d7, d 8, d9, and d10) complexes. For example, the CFSE for high-spin d6 complexes is considered to be ᎑0.4∆o + P (1–3). Bearing in mind that for high-spin complexes P > ∆o, the above expression would lead to the unrealistic result that CFSE is positive (i.e., the system would be less stable for an octahedral field than for a spherical field). The (wrong) inclusion of the pairing energy in the expression CFSE = ᎑0.4∆o + P is due to the fact that two electrons are paired in the same orbital for high-spin d6 complexes (see Fig. 1). However, these electrons would also be paired in the spherical ion. CFSE is the energy difference between the d
Table 1. Crystal Field Stabilization Energies (CFSE) for Octahedral Complexes dn 1
d2
d
Spin State
Configuration
CFSE
–
t2g
1
᎑ 0.4 ∆ o
–
t2g1+1
᎑ 0.8 ∆ o
–
t2g
᎑ 1.2 ∆ o
d4
high low
t2g1+1+1 eg1 t2g2+1+1
᎑ 0.6 ∆ o ᎑ 1.6 ∆ o + P
d5
high low
t2g1+1+1 eg1+1 t2g2+2+1
0 ᎑ 2.0 ∆ o + 2 P
d6
high low
t2g2+1+1 eg1+1 t2g2+2+2
᎑ 0.4 ∆ o ᎑ 2.4 ∆ o + 2 P
d7
high low
t2g2+2+1 eg1+1 t2g2+2+2 eg1
᎑ 0.8 ∆ o ᎑ 1.8 ∆ o + P
d8
–
t2g2+2+2 eg1+1
᎑ 1.2 ∆ o
d
3
1+1+1
2+2+2
2+1
9
–
t2g
d10
–
t2g2+2+2 eg2+2
d
134
eg
᎑ 0.6 ∆ o 0
d4 low spin
d6 high spin
Spherical field
∆o
-1.6 ∆o + P
∆o
-0.4 ∆o
Octahedral field
CFSE
Figure 1. Crystal field splitting and CFSE for octahedral low-spin d4 and high-spin d 6 complexes.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
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electrons in the spherical ion and the d electrons in the crystal field (Fig. 1). Thus, although the pairing energy must be included in the CFSE expression for low-spin d4 complexes, if electrons were paired in the spherical ion as in the highspin d6 example (Fig. 1), there would be no additional pairing energy introduced by the octahedral field. Therefore, the CFSE for high-spin d6 complexes is ᎑0.4∆o , as indicated in Table 1 and in the textbooks by Butler and Harrod (4) and Rodgers (5). It is interesting to note that the excellent textbooks by Huheey contained this kind of mistake in the first two editions (6, 7), and in the latest one (2), but not in the third edition (8). The expressions in Table 1 clearly show that for d4, d5, d6, and d7 metal complexes, the most stable configurations are high spin if P > ∆o, and low spin if ∆o > P. This is not so clear if the average pairing energy is not included in the CFSE expressions.
Literature Cited 1. Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Longman: Essex, U.K., 1992; p 469. 2. Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry, 4th ed.; Harper Collins: New York, 1993; p 400. 3. Cotton, F. A. ; Wilkinson, G.; Gaus, P. L. Basic Inorganic Chemistry, 3rd ed.; Wiley: New York, 1995; p 515. 4. Butler, I. S.; Harrod, J. F. Inorganic Chemistry; BenjaminCummings: Redwood City, CA, 1989; p 395. 5. Rodgers, G. E. Introduction to Coordination, Solid State, and Descriptive Inorganic Chemistry; McGraw-Hill: New York, 1994; p 69. 6. Huheey, J. E. Inorganic Chemistry; Harper and Row: New York, 1972; p 300. 7. Huheey, J. E. Inorganic Chemistry, 2nd ed.; Harper and Row: New York, 1978; p 353. 8. Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper and Row: New York, 1983; p 374.
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