Papers-I
Resource
Prepared under the sponsorship of
The Advisory Council
F.
J. Chem. Educ. 1964.41:466. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 11/22/18. For personal use only.
Cambridge, Mass.
The presentation of ligand field theory in the first college chemistry course cannot be said to be essential. If time is limited, or if the ability of the students and/or their interest in chemistry as a pure science rather than as a branch of useful knowledge are not well above average, I believe there are many other topics which can be more profitably discussed. There is time enough in later years of the chemistry curriculum for those who do elect chemistry as a major subject to tackle this aspect of it. However, in a thorough sort of course, taught to a select, scienceoriented group of students, the subject of ligand field theory might form an interesting and stimulating part. Certainly it is an important part of the “vocabulary” of modern theory which teachers should know so that their presentation not be erroneously simplified at any level. It is important, however, to present it in such a way as to avoid creating false impressions about the nature of ligand-to-metal bonds; this requires of the teacher, first, an awareness and understanding of the dangers and, second, the possession of some concrete ideas about how to avoid them. This article presents an outline of ligand field theory in its present state and at an introductory level; it suggests, partly in precept and partly in example, a presentation of this subject which is primarily intended to be appropriate in the general chemistry course, but it also raises some points which will be of concern to those introducing ligand field theory to students at any level. Ligand field theory can be defined as the theory of (1) the origins and (2) the consequences of the splitting of inner orbitals of ions by their surroundings in chemical compounds. In this article we shall restrict attention primarily to penultimate d orbitals, i.e., the 3d orbitals for ions of the first transition series, 4d and 5d orbitals for ions of the second and third transition series. To a considerable degree, it is possible to deal with the two parts of ligand field theory separately; this has the important consequence that many of the significant and relatively straightforward results of d-orbital splittings, e.g., ligand field stabilization energies, stereochemical preferences and, of course, spectroscopic and magnetic behavior, can be discussed pragmatically without necessarily going very far into the inherently difficult and tedious question of what causes the splittings. Of course, a truly rigorous discussion of all the consequences of inner orbital splittings could not be given without intimately interweaving an examination of the
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College Chemistry
Albert Cotton
Massachusetts Institute of Technology
466
on
Journal of Chemical Education
Field Theory
but for most undergraduate courses—and cerfor the introductory course—such rigor and comtainly causes,
prehensiveness are unnecessary. Causes of Inner Orbital Spittings
The possibility that the degeneracy of atomic orbitals will be significantly split when an ion is placed in a chemical environment was first suggested by Becquerel (1) and the problem was then examined in considerable detail by Bethe {2). Bethe’s work consists of two parts: first, in one of the earliest applications of symmetry arguments to a chemical problem, Bethe determined the qualitative nature of the orbital splittings for various important geometries. These qualitative results are correct whatever the mechanism
(electrostatic, or covalent, as discussed presently) which brings them about. For d orbitals, in several important geometrical situations, they are as follows: In both octahedral and tetrahedral surroundings, the dZVl and dvz orbitals remain equivalent, as do dx*-yz and dzs orbitals. In square surroundings, the dzi and dvz orbitals remain equivalent, but the dzy, dxi~yz and dzt orbitals are not equivalent to any dzz,
others.
Bethe obtained these and other results by formal, group-theoretical methods, but they can also be obtained, or at least their correctness strongly suggested, “Resource Papers” is a series being prepared under the sponsorship of the Advisory Council on College Chemistry as one of the activities of the Teaching Aids Panel. The Advisory Council on College Chemistry (AC3) is supported by the National Science Foundation. Professor Charles C. Price, of the University of Pennsylvania, Philadelphia, Penna. 19104, is the chairman. Single copy reprints of this paper are being sent to chemistry department chairmen of every U.S. institution offering college chemistry courses and to others on the mailing list for the ACS Newsletter. Additional single copies will be sent free to all interested individuals who make request to the Editor of the ACS Newsletter: Professor E, L. Haenisch
Department of Chemistry Wabash College
Crawfordsville, Indiana 47933 Multiple copy orders (in lots of. 10) can be filled if accompanied by remittance of $1,50 per unit of 10 copies. Orders must be addressed to Professor Haenisch, not to the
Journal of Chemical Education.
by informal, pictorial arguments, requiring only the To common sense* any bright freshman should have. do so, it is first necessary to present pictures showing the shapes of the usual five d orbitals, Figure 1. In addition, it should be pointed out that the de« orbital can be regarded as a combination, in equal parts, of two orbitals, dst-xi and dz*~y*, each of which is shaped like the other four d orbitals; this is illustrated in Figure 2.2 It is now supposed that the metal ion is placed in an octahedral array of six ligands, as shown in Figure 3. 1
a tetrahedral set of four ligands, arshown in Figure 4 at alternate vertices of a cube,3 it can be seen that the dxy, dxz, and dyz orbitals stand in one sort of relationship to the ligands, namely, with their lobes pointing to cube edges, while the
Similarly, for
ranged
as
The arrangement of the six in relation to the same Cartesian nd 2 for the d orbitals.
Figure 3.
plex
ligand atoms in an octahedral comcoordinate system used in Figures
1
Balloon pictures of the conventional set of d orbitals. The surdrawn to enclose a major pari—say, roughly, 90%—of the amplitude of the wave functions. The sign of the wave functions in each lobe is shown. Since the distribution of electron density is given by \p2, it will in each case be quite similar to the shape of the wave functions of the orbitals.
Figuie 1. faces are
Figure 4. The arrangement of the four ligand atoms in a tetrhaedral complex in relation to the same Cartesian coordinate system used in Figures 1 and 2. Note how the tetrahedron is composed of four alternate corners of a cube.
Z
Drawings showing how the dz2 orbital consists of dj2_v2 orbital in equal proportions.
Figure 2. a
a
d32—z2
and
It should not be hard to see that the dX2-y%, dzi-xi, and dz* V2 orbitals are all oriented in one way in relation to the six ligands, while the dxy, dxz, and dyz orbitals are all oriented in a second way. Specifically, all those in the first set have each of their lobes going toward a ligand atom, while each one in the second set has each lobe going between ligands. Thus, we get the result mentioned above that the dxy, dxz, and dyz orbitals are equivalent to one another, whereas the dxi~yt and dzt (being made up of equal parts of the equivalent pair dzt-xdzi-„z) are different from the first three, but equivalent to one another. -
1
is
Someone ia supposed to have remarked that “group theory common sense,” which is not too great an
just organized
exaggeration. 1 For algebraic details, Bee Cotton and Wilkinson, page 973i It is necessary to make this breakdown of the dz* orbital in order to demonstrate by the pictorial argument that dz* and dx* y* are not only different from the other three, but also are equivalent to one another. It is to be stressed, however, that the dzi—x* and dz2-y* orbitals have no actual existence along with the other four, since there can be only five independent nd wave functions. —
Figure 5. The arrangement of four ligand atoms in a square complex in relation to the same Cartesian coordinate system used in Figures 1 and 2.
dxz-yi, dzz^xZ, and dzi-yi orbitals stand in another relationship; namely, their lobes point to the centers of cube faces. Hence, again, we conclude that dxv, dxz, dyz form one equivalent set while dxZ^y2 and dz* form a second equivalent set. Finally, it can be seen that the dxz and dyz orbitals both have the same relationship to the ligands set at the of a square in Figure 5, that both dzt-xt and corners The usefulness of this way of looking at a tetrahedron, for many purposes besides the present one, is worth emphasizing. Volume 41, Number 9, September 1964
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467
d^-y2 have a second relationship to the ligands and that each of the two remaining orbitals is related in still different ways. The question now arises as to how much the energies of the nonequivalent orbitals, or sets of orbitals, may differ. It must be assumed that if two orbitals, even if intrinsically similar, are differently oriented toward their surroundings, there will, in principle, be some difference in the energy of an electron depending on which of them it occupies. The second part of Bethe’s paper offered an answer to this question based on the assumption that the ligands can be treated as point negative charges, a model which has often been considered to represent the chemist’s concept of purely ionic bonding. (In fact, it does not, as will be seen.) Using this model, it is easy to see that qualitatively, an electron in an orbital whose lobes point at the negatively charged ligands will have a higher electrostatic potential energy than an electron in an orbital whose lobes point between ligands. In other words, the former orbital is a less stable one and lies higher on an energy level diagram of the usual type than does the latter orbital. Figure 6 illustrates these results for the octahedral, tetrahedral and planar cases. It should be noted that for the square environment, the ordering of the orbitals is not exactly fixed by the symmetry alone, but depends in certain respects (e.g., as to whether the d# orbital lies above or below the dxy orbital) on the physical details. The arrangement shown in Figure 6 is the one believed to be correct in most real cases. Move generally, the relative stabilities of a series of inherently similar orbitals will vary inversely according to the extent to which they bring the electron into proximity with the negative ligands. The argument may be generalized to include not only negatively charged ligands, but dipolar ones, which will be so oriented as to have their negative ends closer than their positive ends to the metal ion. Bethe showed further that the actual magnitudes of the energy differences (e.g., A0 and A, in Figure 6) can be calculated using this electrostatic model if one can select the proper magnitudes for the metal-ligand distance, and the ligand charge (or dipole moment) and assign an appropriate radial part4 to the wave function for the d orbitals. In relatively recent times, Ballhausen (3) showed that if these parameters arc assumed to have the same values in an octahedral complex, MXe and a corresponding tetrahedral one, MXf, the ratio of A; to A0 will be 4/9. It has also been shown that the absolute values of A0 (or A,) can be obtained within a factor of less than two by choosing reasonable values for the various parameters required. However, in spite of all this sort of pragmatic success, this electrostatic model is now known to be unrealistic and ultimately unsatisfactory. It has been discredited both experimentally and theoretically. The experimental evidence against it comes from a variety of sources all showing that electrons which are supposed to be entirely in the metal ion d orbitals, according to * When an orbital wave function is expressed in spherical coordinates ip(r, 0, ) R(r)Q{$)®{), the radial part is the R{r). The parameter r measures the radial distance from the origin. The angular part of the wave function is ©(0)(0) which involves the direction in space from the origin and depends only upon the angles 0 and 0 NH3
~
~
many other oxygen Uganda < many nitrogen ligands < en
