Arthur W. Adamson University of Southern Colifornio Los Angeles
Domain Representations of Orbitals
There has been a great development in recent years in the use of qualitative wave mechanics in the de~cript~ion of chemical bonding to high school and to undergraduate college students. The terms sigma and pi bonding have percolated quite far down the pedagogical tree as have wave mechanical explanations of bond angles and molecular geometry in general. Even ligand field theory or a t least the more simple crystal field arguments are beginning to appear familiar a t these levels. In order to accomplish such a tour de force, very effective use is made of simple pictorial and model representations of hydrogen-like orbital functions. One can, in fact, h d quite a variety of such depictions. The $ function for a p-orbital may be shown as a pair of tangent spheres (or circles in a projection) either sharp or fuzzy in outline, and possibly signed so as to allow qualitative discussion of overlap integrals. The function map alternatively he shown as a pair of prolate ellipsoids, now related to $2 rather than to $; they may appear as detached tear drops, suggesting a contour of constant electron probability. Some of these forms are illustrated in Figure 1.
concern (3). Even in senior level college texts it often is simply not mentioned that the lobe diagrams are merely polar plots of the trigonometric portion of the wave function, and thus constitute a showing of the angular variation of electron distribution a t a constant radius. That is, distance from the center in a lobe plot is not distance from the nucleus, but is a quantity proportional to the value of the trigonometric function.
Figure 2. d-Orbital.. lo1 Polar plot ofY 5%(dil,d in xy plone. (b) Polar plot of Y22zld.,) in xy plone. Icl Polor plot of YPzdd,? in x z plone.
Thus for the 4p-orbitals, important for first row transition elements, the functions are (4): *(p,) = R(r)(3/4r)'/asin 8 cos B $(p,)
=
R(r)(3/4n)'/*sin e sin 4
+ ( p a )= R(r)(3/4rr)'lacos 0 a
b
Figure 1. V o r i o u ways of rhowing p-orbitals. (ol Polar plots of YI.,I in the xy plane. (b) Polar plots of Y',.,, in the xy plone. (c) Qualitative irodenrity contours of P, and P, orbitals.
Similarly, the set of &orbitals have acquired a certain familiarity. We have the dij functions, with their four lobes pointing away from the rectangular coordinate axes, the d,.-+ orbital, identical to d,, except for orientation, and the odd-shaped dB*one. A typical set of projections is illustrated in Figure 2. Even f-orbitals have started to become more familiar; projection and perspective drawings have recently been published in THIS JOURNAL. (1, 2 ) . The familiarity of g-orbitals will perhaps he longer in coming. While these lobe representations of orbitals serve an excellent teaching function, the fact that they are so often used for qualitative theorizing about bonding by students who do not do actual calculations can lead to an over literal acceptance of lobes as a kind of photograph of an atom. Others have expressed the same 140
/ Journol of Chemicol Education
and Z is the (effective)nuclear chargenandao,the radius of the first Bohr orbit (a0 = 0.53 A). The angles B and are for the polar coordinate system shown in Figure 3. The normalized angular portious of the $ functions will hereafter he abbreviated xJ/,/ I Y I , The plots of FigY are l a are just the polar plots of sin 0 cos and of sin 8 sin @, and those of Figure 2a, their squares. ~i~~~~ 3. polar coordinate The next group of or-
+
I
+
bilals of importance in thc considcrat~onof first row transition elr~nents is the srt of :id-orbitals fnoctions: +(d,?)
=
R'(r)(5/1G~)'/r ( 3 cusx 8 - 1)
(5)
+(dis~ys) = R'(r)(15/16~)'/% sinz 6 cos 24
(6)
+(d,,)
=
R'(r)(15/16s)lh
(7)
J.(d,.)
=
K1(7)(15/4a)'/?sin 6 aw 6 crm 4
(8)
( 1 )
=
11'(r)(15/4~)'/?sin 8 ens 8 sill
+
(9)
sill2 8 sin
.+domain ! is dejined as that region in space in which the elect~on(lensil?/for a particular orbital i s greater than that ,for an%/other i~iwnberof th,e set.
24
The plots in Figure 2 are simply those of (3 cos28 for d2,,, (cos 2~75)~ for d , . ~ , in ~ thc xy plane, and (sin 24)2 for d, in the xu plane. Thc lobe plols, iu spit,e of their u~iquestio~lcd usefulness, do obscure or distort several importaut aspects of the con~plete functions. It has been poi~iledout t.hat if constant electron probability contours are drawo, lhcn inclusion of the radial port,ion of the 2pfunctions gives a inore flattened appearance to t,he lobes (6). Furthcr, if t,he lobe plot,s are treated as representations of t,he physical extension in space of electron densit,y, onc is ignoring the fact that Lherc may be nodes in thc radial porlion of the co1nplet.e function. Thus for t,hc 4p set, there arc two nodes (or values of r a t which = 0 (the solotions t,o the quadratic portion of R(T)in equation (4), while there are none for thc 3d set,. The lobe plots also cnt,irely obscure the fact that thc sct of orbitals of a given 1 value are spherically synunetric. Thus fi2(p,) +2(p,) +"p.) = constant (6). An important consequence is that the filled p or d sub-shcll has no net angular n ~ o l ~ ~ e o t u m The . point that thc group of funrt,iol~s
The contrast between the lobe aud domain approaches is illuslraled in Figure 4, which shows the two types of plots in the rase of +2(p,) and +2(p,) in the xy plane.
+
+
+
+
constitutes a set of spherical harn~onicsaud essentially a way of portioning a sphere is a useful on?, {.hen; it should somehow he ~ossiblet,o bring this out in a simple model. Finally, in the case of t.he d-orbit,al set, the special appearance of dn2gives the impression that the atom possesses a unique axis, although reflection must assure one that this cannot be so in the abscnce of external fields. Evcu in a crystal field, if the synuuel,ry is octahedral, lhcre is I I O way of deciding that a particular cartcsian axis is the z axis aud the home of the dzZo r b 1,aI.
Figure 4. Lobe verrvr domoin representation of p-orbitals in the xy plone. in the xy plone. Ibl Domain representation. Poior plots of YI.
(01
The former, shown in Figure 4a, consists of the polar plots of cos2+ and sin2+, respectively, while the latter shows, out t,o a radius r in real space, the regions of dominance of the two functions. The domain boundaries are simply straight lines, since they represent solutions for the condition:
or, since the radial and constant port.iou are the same, for
Exccpt for labeling, the plols in the xz and ?jz planes are the same. The three dimensional model of the domain representation is shown in Figure 5. Each p-orbital has two domains in the form of oppositely lying sectors of a sphere; the boundary of each is a spherical square. This last feature, a t first surprising and even disturbing lo the person entirely accustomed to seeing cylindrically syin~netriclobcs, occurs because the domain procedure is essentially one of dividing a sphere into sectors, in t,his case, six equivalent ones.
Orbital Domains
1:or son~otilnc ih?re has b r m a defiuite nccd for somc other way of displaying the various orhital functionsa may in which distanrc from t,he origin correspouds t.o act,nal distarre in spare fro111t,lrc ~ ~ n r l e uand s , a way which lnalxx morc apparel11 lhr features obscured 1)y the lobe represent.atior~s. I n fact lhc existence of so Irrauy different, ways in which attrnipts are made lo show thc lobc t.ype diagrams is probably ail indioat.ion of a general frcling that such diagranis are not fully adrquatc. The allcrnalivr may which is proposed of what mill he called o&tal dohere is the dcpictio~~ 17iain8.
Figure 5.
Model of p-orhitoi
domains.
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3, March 1965
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141
No essential symnletry has been lost in the domain model, however. The set has the Oh symmetry (7) of the set of three p-orbitals. Bonding argunlenls are unchanged. Thus, as is shown in Figure 6, sigma honding can be illustrated by having two of the models approach each other along one of their axes. The domains overlap symmetrically, and the requirement for a sigma bond that t,here be no nodal plane is met.
Figure 8. Lobe versus domain reprerentations of d-orbifals. ( 0 ) Polor pioh of Y Z functions ford,. 0 n d d . z _ ~ 2 i n thexr plone. Ib)Corresponding domain representation.
Figure 6. Use of domoin representotions to show bonding. Upper Lower drawing: projection view of two drawing: in xy solid models in contact.
Again referring to Figure 6, a bonding would be due to overlap of p, and/or p, orbitals, and it now becomes quite apparent why sigma bonding takes precedence over a bonding. Very considerable overlap must occur in the p, domains before much is present in the a bonding overlap. A small point is that the square contour of the abutted domains that overlap for sigma bonding carries the correct implicat,ionthat the nlolecule is not invariant in energy touvard rotation of one at,om relative to the other on the bonding axis. This is not a consequence of any physical reality to the shape of the domain boundaries, but merely of the fact that the symmetry of the set of domains is that of the set of orbitals themselves. The domain approach focuses attention on a set of orbitals in terms of what they are, namely a set of spherical harmonics. I t is then easy to see that a sphere can be divided in more than one manner. Another way is shown in Figure 7, which is a picture
of a model for the domains of a set of four tetrahedral hybrid orbitals. Each domain boundary is now a spherical triangle; the model has T d symmetry. As with the previous case, the domain boundaries are planar, i.e., angularly independent of dist,ance away from the nucleus. This characteristic is independent of the nature of R(r) since the function is the same for each orbit,al. Again bonding can be illustrated; in fact, the large solid angle subtended by each domain assists in making plausible the point that the amount of overlap and hence bond strength is not very sensitive to small depart,ures from the t,etrahedral angles. Turning next to d-orbitals, t,he lobe and domain presentations are illustrated for the s y and zz planes in Figures 8 and 9, respectively. The problem, in the case of the lobe diagrams, of showing two or more orbitals in the same drawing and hence in relation to each other, is now acute because of the overlapping. The spacial arrangement of the lobes can be visualized as follows: the d,% and d,.-,. provide together six lobes point.ing t,omard the six faces of a cube, while each d,. orbital has four lobes pointing toward the centers of four edges of a cube. Thus the dzForbkal has four lobes pointing to the centers of those edges which are perpendicular to the s y plane. Again, the problem encount,ered in trying to show this in a threedimensional model is that the lobes overlap. The domains,
a
b
Figure 9. Lobe versus domoin representolion. of d-orbitals. lo) Pdor plob of YPfunctionr for d d , dn, and d.2-,l in x z plone. (bl Corresponding Figure 7.
142
/
Model of rpQhybrid orbit01 domains.
Journal of Chemical Educofion
domain representation.
however, by defiuition do not do so. Note that the boundaries of the domain sectors occur a t the angles at which, in the lobe drawings, there is a crossing of the polar plots. Thus the d,=-,. - d,, boundary is a t = 22' 30' (and at the three angles generated by the CI axis (7)) and the d.. - d,, and d,, - d,.-,z boundaries are at 8 = 26'45' and 0 = 63" 46' respectively (and a t the other angles generated by symmetry). I t will be noticed that the d.. and d,2-,2 domains are not equivalent and, in fact, that the domain plots in the xy and xz planes are not equivalent. I n obedience to this asyn~n~etry the boundary of the d,.-,. domain, in the solid model, is not a spherical square and does not show a fourfold axis. Since most qualitative discussions of the arrangements of d-orbitals in space are for the isolated atom or for octahedrally bonded atoms, it seems desirable to form a combination of the orbitals which does not make one axis unique and which has the desired symmetry. First, the d,. set is already satisfactory; it is symmetric with respect to the three axes. However, if the three axes are indistinguishable, the remaining two orbitals could equally well be written as
+
d*O, (d.l.l, dy'), or ( d " l 2 , d.2)
If the average of the above three equivalent designations is taken, a new psi fnnction is obtained:
d.,), and three sets of four domains each (those for d,,, d,., and d,.); each of these last has the shape of a flattened hexagon. The polyhedron corresponding to the model is a truncated rhombohedra1 dodecahedron. The volumes of the domains are very close to simple expectation; that is, four times the volume of one hexagonal domain is within 5y0 of being equal to six times the volun~eof one square domain. The symmetry of the model is Or, and as with t,he p-orbital set, no essential features of symmetry or of ability to discuss bonding are lost relative to t,he lobe model. It is easy to see that in an octahedral field, d-orbitals fall into a set of three (the dij) domains, and the double d,, domain (8). Simple crystal field arguments are t,hus easy to rat,ionalize. Octahedral complexes clearly can use the d,, set for sigma bonding, with the djj set in position for a bonding. In tetrahedral geometry, one observes that t,he points of juncture of three dij domains lie a t the corners of the cube. Alternate junctures are thus tetrahedrally disposed and able to participate in sigma bonding. The square domains are in position for T bonding. f-Orbitals
Turning finally to the case of f-orbitals, the set hybridized for cubic symmetry is (1,3,9) (angular portion only) :
This reduces to: +2(e,)
=
R'(r)2(5/r)(1/8)[1
- 3 (x2g2+ xZzZ+ y?Z2)1 (13)
where x, y, and z designate the corresponding projections of a unit vect,or. J.(e,) is seen to be a normalized function which is symmetric with respect to the three axes, and on replacing x, y, and z by their trigonometric equivalents, one obt,ains +'(e,)
=
(5/r)(1/8)[1
- 3 sin% (sin2E cos2+ sinP+ + cos28)]
fSr,?-,.,
=
(14)
It perhaps should be emphasized that J.(e,) is not a hybrid orbital sirice it is a combinat,ion of #2 rather than of # functions. However, it is not intended for use in calculations but only as a device for showing the average electron density of e,-orbitals if no one axis is special. The domain boundaries obtained on using the $P(dij) functions (Equations 7, 8, and 9) and the fi2(d,,) function are shown for the xy and xz planes in Figure 10; these projections are now the same. The full appearance of the domains is shown in Figure 11. The solid model displays six square domains (t,hose for
Figure 10. d-Orbltol$ domoinr "ring plane. (b) In the xr plane.
fvi,l_,~,
=
!dEsin 8 eos 6 (cus9 - sinP8sin9+)
(19)
ap
(20)
4
r
r
sin 8 sin + (eost8
- sinz8 cosP+)
The L,,-orbital, in a polar plot, gives eight equivalent lobes pointing toward the rorners of a cube. The f,$-orbital gives two opposite lobes along the x-axis, and
d., [Eqvotion 141. (o) In the xy Figure 11.
Model of d-orbital domoinr.
Volume 42, Number 3 , Morch 1965
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143
hence centered on opposite faces of the cube, and the other two functions similarly produce lobes along the y and z axes, so that the triply degenerat,e set yields six equivalent lobcs centered on the six sides of a cube. Finally, t,he j,(..-,.)-orbital gives eight equivalent lobes centered on eight of the twelve edges of the cube (the four edges not involved being those whose centers lie in the yz plane). The other two orbitals sin~ilarly give eight lobes each. For the purpose of obtaining domains, it is necessary to perform a similar kind of averaging on this last set as was done for the d,s-,., d,. one, that is to define a new f.as the average over all permutations of the labelling of the axes, i.e.. of the orbitals. One thus fz(.s-r'), j Z y ( z ~ - z a ) , and f.c,~-,*) obtains
The average function f, gives twelve lobes pointing to the cent,ers of the twelve edges of the cube.
relation to each other-a difficult feat if lobe diagrams are used. The solid model is shown in Figure 13. Summary
The domain approach to the display of various sets of orbitals is suggested as one which helps to clarify aspects that are difficult to see in terms of lobe diagrams. I t is particularly useful as a means of s h o ~ ~ i nthe g geometric relationships of a set of orbitals and hence should be useful in illustrating various features of molecular geometry and bonding. I t perhaps should he emphasized that the domain type of model is offered as complimentary to (not in opposition to) lobe diagrams. Use of the two together should considerably facilitat,e the development of a good qualitative understanding on the part of students of the wave-mechanical approach to chemical bonding. Construction of Models
I t may be of interest to the reader to describe how the models shown were constructed. The general procedure was the following. First, an accurate sector corresponding to each type of domain was machined in aluminum. I n each case, a billet was turned on a lathe so that the end had a spherical surface of desired radius. A hole was drilled and taped in the center of the end, so that a screw could be inserted which would allow the piece to be held in the chuck of a milling machine. The piece was then milled at the various appropriate angles to give the flat sides of each b domain; the sides, o f course, came together at the Figure 12. Polyhedra corresponding to (-1 d-orbital domains and (bl point corresponding to the center of the sphere. f-orbital domainr. The angles involved for the p-orbital and the tetraThe domain model will thus consist of eight f,,,or hedral hybrid orbit set are fairly obvious, but those for f a domains, two domains each from f,., f and f-,*the d-orbital domains are perhaps not. For the square which form a set of six equivalent doniains which it is domain, four cuts are t,aken on the milling machine, a t convenient to label the fa domains, and the set of 90' to each other, and each at an angle of 20' 54' to the twelve f,domains. It turns out that the twelve B axis of the domain. I n the case of t,he hexagonal domains are rectangular in shape, the eight 13 domains sector, the first milled surface is at an angle of 24%' to hexagonal, and the six 6 domains octahedral. A the axis of the sector. For the second cut, the sector is picture of the corresponding polyhedron is shown in rotated 57'36', and the cut is made at an angle of 30' Figure 12. The figure can be thought of as derived t,o the axis. For the next cut, the rotation is 64'38', from the d-orbital domain polyhedron by a truncation and the angle is again 30'; for the next, the rotation is of each of the points a t which three hexagonal faces 57'36', and the angle is 24'6'; for the next, the meet; alternatively, it is related to the Archimedian rotation is 57'36' and the angle is 30°, and, finally, for truncated cuboctahedron. This domain model has the last. side, the rotation is 64'48' and the angle is 30°. Oh symmetry; it shows the entire set of f-orbitals in The cuts were made in small increments until finally all sides met at a point. The metal sect,ors or doniains mere then used to form a ceramic mold, which was fired xith a light glaze. The sectors for the actual models were then cast in colored, transparent polyester plastic. They held together by means of a center piece which consists of a steel cube drilled and taped at the appropriate angles so that pins could be attached. Each plastic sector was sufficient,ly truncated to allow space for the center piece, and the pin inserts in a hole drilled in the truncated end, on the sector axis. 0
Acknowledgments
Figure 13.
144
Model of f.0rbit.d domain..
/ Journal o f Chemical Education
The writer is greatly iudebted to Professor C. F. Ball of the Ceramics Department for assistance in the ~reparat~ion of the molds, to H. Gray and J. Donohue
for helpful discussions on the geometric figures involved, and to Miss Reinecke for construction of cardboard models of the d- and f-orbital domain polyhedra. Literature Cited (1) FRIEDMAN, H. G., JR., AND FEUERBACAER, D. G., J. CHEM. Eouc., 41, 354 (1964). (2) B E C ~ E R C., , J. CBEM.EDUC.41, 358 (1964). (3) OGRYZLO, E. A., AND PORTER,G. B., J. CHEM.EDUC.,40,256 (1963).
(4) PAULING, L., AND WILSON,E. B., "Introductio~~ to Q ~ ~ a n t u m Mechanics," McGraw-Hill Book Co., New York, 1935, p. 134. (5) COHEN,I., J. CHEM.EDUC.,38, 20 (1961). (6) U ~ s o m A,, , Ann. d. Phys., 82, 355 (1927). (7) For a brief discussion of symmetry groups and operations,
see ADAMSON, A. W., "Understanding Physical Chemistry," Part 11, W. A. Benjamin Book Co., New York, 1964.
(8) COTTON,F. A., J. CHEM.EDUC.,41,466 (1964). (9) GRAY,H., private communication.
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