H. G. Friedmon, Jr.,' G. R. Choppin, and D. G. Feuerbocherz
Florida State University Tallahassee
(
The Shops of the
f Orbitals
I
The shapes of the s, p, and d orbitals are familiar to all students of modern chemistry, while the shapes of the f orbitals are relatively unknown. Although the equations for the f orbitals are readily available, meaningful illustrations are difficultto find, either in research journals or in chemistry textbooks. A few advanced texts (e.g., that by Pauling and Wilson (1)) present plots of the 0 part of the f wave functions. However, a t best, these are only cross sections of the f orbitals, and are not necessarily good indications of the overall shapes. In this paper, we have attempted to present illustrations of the shapes of f orbitals which may he of interest to students of inorganic chemistry. One reason for the neglect of the f orbitals is the generally accepted belief that they have little or no effect on honding. Although the f orbitals certainly do not play as important a role in bonding as do the d orbitals, their involvement may account for some of the more subtle aspects of the chemistry of the elements with atomic number 51 to 70 and 83 to 102. Hugus in an interesting paper (2) several years ago suggested that the 4f orbitals may be involved in bonding to some extent in the elements a few atomic numbers lighter than lanthanum. The enhanced stability of the higher oxidation states of antimony, tellurium, and iodine relative to those of arsenic, selenium, and bromine, respectively, was attributed to the contribution of the 4f orbitals to the total bonding In addition, it has been suggested (5) that the 4forbitals in the lanthanide elementsmay he split incomplexes by ligand fields, resulting in as much as a 10% increase in complex stability in the most favorable cases. This latter suggestion has not yet definitely been verified by experiment although there is inconclusive evidence (4) to support it. The 5f orbitals of the actinide elements seem to he available for participation in bonding, a t least up to plutonium. Thus, the stability of the M02++ and M02+ions (i\4 = U, Np, or Pn) has been attributed (5) to participation of 5f orbitals in bond formation. A distinct difference in behavior toward ion exchange resins between the lanthanide ions and the transplutonium actinide ions in concentrated hydrochloric acid solution has been explained, also, by greater involvement of the 5f orbitals in bonding (6). The f orbital pictures in this paper are based on the angular part of the hydrogenic wave functions. These are found, for example, in Appendix I11 of the third
edition of The Nature of the Chemical Bond (7), and in a paper by J. C. Eisenstein (8). (Pauling and Eisenstein used different normalizing factors, resulting in equations that differ by the factor 6.) The orbitals are usually represented in a spherical coordinate system, as shown in Figure 1. In this coordinate system, the total wave function, J., can he expressed as the product of three functions, each of which is dependent only on one coordinate:
IPresent address: Defensive Research Division, Chemical Research and Development Laboratories, Edgewood Arsenal, Xaryland. Present address: Dept. of Chemistry, University of Houston, Houston, Texas.
The f orbitals are unusual in that no single set is equally useful in all problems. One set, which me shall call the general set, is obtained in the usual way, i.e., by forming the product O(9) .a(+). However, this set con-
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+.rm
2
f
Y
x
I Figure 1. The Carterion coordinoter ( x , Y , zl ond the gherical coordinoter (1.8,01 of o point P.
(7,
8, +) = Rnr(r).Brn (E).%(+) (1)
where n is the principle quantum number, 1 is the azimuthal quantum number, and m is the magnetic quantum number. The shapes of the orbitals are determined primarily by the angular part of J.. Y,, (8,m)
=
elm (8).lpm (6) (2)
Therefore, we are concerned principally with equations for Y in terms of 0 and 6 Each wave function must he normalized for comparison with other wave functions. This means that the constants in the expression for J. must be adjusted so that f +*+ dv
=
normdilization factor
(3)
where du = rz sin 0 d+ d0 dr is the differential of volume, and the integration is taken over all space. J.* is the complex conjugate of J.. In the case of real wave functions, such as those in this paper, J.* = J., and J.*+ = . The normalization factor is usually taken as 1 (e.g., Pauling) when the total wave function is being considered, or as 4 s (e.g., Eisenstein) when only the angular part is being considered. In this work, the angular part and the radial part of the wave functions have been separately normalized to 1; i.e.,
scale. The sign of the wave function in each lobe is indicated on the cross sections. For clarity, the signs are not shown on the three dimensional projections, but may be obtained by inspection of the cross sections and of the polynon~ialsubscripts. (Adjacent lobes of the same orbital always have different signs.) Figure 2 shows a three-dimensional view of the f,s orbital, with a section cut out of the two "collars." Figure 3 is a cross section of the same orbital in any plane containing the z-axis. The fZ3 and f,. orbitals are identical, except that they extend along the x- and yaxes, respectively.
1 Figure 4.
The fdorbitol.
Figure 6.
The f,i,z.8,*,
Figwe 5. A cross redion of the fr,%arbitol in the xz-plane.
Figure 7. A cross section of the fzrs?au~i orbit01 in the xy-plone.
orbital.
Figure 4 is a thrce-dinlensional view of the fZx2 orhital, and Figure 5 is the cross section of this orbital in the xz-plane. The f,,, orbital is identical in shape, but is rotated 90" around the zaxis, so that the axis of the small lobes (the horiFigure 8. The f-, orbital. zontal axis in Fiu. 51, is the y-axis. The cross section of each lobe of these orbitals, in a plane perpendicular to that of Figure 5 and passing through thc nucleus, is a circle. As a result, these lobes appear flattened, resembling a h a bean. A three-dimensional view of the f , , , ~ , , ~ ~orbital is shown in Figure 6, and a cross sectional view (the zyplane) in Figure 7. The f,,,,~,~, orbital is the same shape, being formed by rotating the lohes of Figures 6 and 7 about the z-axis 90' clockwise. All six lohes of each orbital are identical. The cross section of each lobe in any plane perpendicular to the axis of the lobe is elliptical. The f,,, orbital, shown in Figure 8, has eight identical lohes which point toward the corners of a cube whose center is at the nucleus. The f,,,.-,a,, f,,+,aI, and
orbitals are the same shape, but are rotated 4 5 O about the x-, y-, and z-axes, respectively.
f,,,L,.l
Ligand Field Splitting o f f Orbitals
It was stated earlier that group theory predicts that in a cubic-type ligand field, the f orbitals should split into two groups of triply-degenerate orbitals and one nondegenerate orbital. I
